INTERNATIONAL GONGRESS , thus the scheme-theoretical intersection Spec(Q_(delta_(c)^(2omega_(1))))\operatorname{Spec}\left(Q_{\delta_{c}^{2 \omega_{1}}}\right) of the upward flow W_(delta_(c)^(2omega_(1)))^(+)W_{\delta_{c}^{2 \omega_{1}}}^{+}and the nilpotent cone h^(-1)(0)h^{-1}(0) is the line (a_(0))\left(a_{0}\right) with a double embedded point at the origin. Note that this upward flow was studied in [29, $8.2].
Both (3.10) and (3.11) follow from Conjecture 3.12.4, and both can be proved by direct computation in Gr^(mu)\mathrm{Gr}^{\mu} as explained above.
Remark 3.15. It is surprising how complex J_(d)(Spec(H^(**)(Gr(k,n),C)))J_{d}\left(\operatorname{Spec}\left(H^{*}(\operatorname{Gr}(k, n), \mathbb{C})\right)\right) can be. In particular, in the k=1k=1 case (i.e., jet schemes of the cohomology ring of projective space) there is only a conjecture about its multiplicity in [50, CONJECTURE III.21].
Remark 3.16. Finally, we remark that already for type (2) we have new phenomena. As discussed in [37,$5.4][37, \$ 5.4], there are multiplicity algebras depending on continuous parameters, in particular they cannot be isomorphic to cohomology rings, because cohomology rings are integral.
3.3.1. Lagrangian closure of W_(delta)^(+)W_{\delta}^{+}
Definition 3.17. Let EinM^(s pi)\mathscr{E} \in \mathcal{M}^{s \pi}. The Lagrangian closure bar(bar(W_(E)^(+)))\overline{\overline{W_{\mathcal{E}}^{+}}}of W_(E)^(+)W_{\mathcal{E}}^{+}is the smallest closed union of upward flows containing W_(E)^(+)W_{\mathcal{E}}^{+}. In other words, the Lagrangian closure is the closure in the quotient space by the BB partition.
Using (3.8) and (3.4), we can deduce the following
Theorem 3.18 ([30]). Let mu inP^(+)\mu \in P^{+}and c in Cc \in C. Recall delta_(c)^(mu)\delta_{c}^{\mu} from (3.7). Assume E_(delta_(c)^(mu))inM^(s pi)E_{\delta_{c}^{\mu}} \in \mathcal{M}^{s \pi}. Then
i.e., the upward flows correspond to dominant weights lambda\lambda less than or equal to mu\mu in dominance order.
3.4. Towards a classical limit of the geometric Satake correspondence
Finally, we will formulate some conjectures which were the original motivation of much of the previous ideas. In particular, they hint at a new construction of the irreducible representations of GL_(n)(C)\mathrm{GL}_{n}(\mathbb{C}), and more generally of any complex reductive group G\mathrm{G}.
The general setup comes from the classical limit (2.5) of the geometric Langlands program, as formulated in [13]. Here we sketch some of the expectations of this classical limit in a schematic (not completely well defined) manner. It should be an equivalence of some sort of derived categories of coherent sheaves
Several properties of this equivalence were proposed and some established in [13]. In particular, S\mathscr{S} should be a relative Fourier-Mukai transform along the generic locus in A_(G)~=A_(G)\mathcal{A}_{\mathrm{G}} \cong \mathscr{A}_{\mathrm{G}}. Another crucial property [38], which we called enhanced mirror symmetry in Section 2.5 above, is that S\mathscr{S} should intertwine the actions of certain Hecke operators on D^(b)(M_("Dol "))D^{b}\left(\mathcal{M}_{\text {Dol }}\right) and the Wilson operators on D^(b)(M_("Dol "))D^{b}\left(\mathcal{M}_{\text {Dol }}\right). Let mu inX_(+)(G^(vv))=X^(+)(G)\mu \in X_{+}\left(\mathrm{G}^{\vee}\right)=X^{+}(\mathrm{G}) be a dominant character of G^(vv)\mathrm{G}^{\vee}. We denote by
some space of Hecke correspondences at a point c in Cc \in C. Indeed, this gives us
two maps to M_("Dol ")\mathcal{M}_{\text {Dol }}, first the projection pi_(mu)\pi_{\mu} to the first factor, and second f^(mu)f^{\mu}, the Hecke transformation ^(1){ }^{1} of (E,Phi)(E, \Phi) by the compatible Hecke transform [g]inGr^(mu)[g] \in \mathrm{Gr}^{\mu}, which is expected to induce
given by tensoring with the universal G^(vv)\mathrm{G}^{\vee} bundle E\mathbb{E} in the representation rho_(mu):G^(vv)rarrGL(V_(rho_(mu)))\rho_{\mu}: \mathrm{G}^{\vee} \rightarrow \mathrm{GL}\left(V_{\rho_{\mu}}\right).
There are two more expectations for the classical limit S\mathscr{S}, both are motivated from Fourier-Mukai transform where the analogous statements hold. First, we expect that for any FinD^(b)(M_("Dol "))\mathscr{F} \in D^{b}\left(\mathcal{M}_{\text {Dol }}\right) we should have
where W_(mu)^(+)W_{\mu}^{+}is the upward flow from a certain epsi_(mu)\varepsilon_{\mu} maximally split GG-Higgs bundle of type mu\mu at c in Cc \in C. On the other hand,
This follows from Theorem 3.6 and a direct computation for chi_(T)(E_(c))\chi_{\mathbb{T}}\left(\mathbb{E}_{c}\right).
In [29,$8.2][29, \$ 8.2] we proposed that for n=2n=2 the mirror of Sym^(2)(E_(c))\operatorname{Sym}^{2}\left(\mathbb{E}_{c}\right) should be the struc-
ture sheaf of the Lagrangian closure bar(bar(W_(delta_(c)^(2))^(+)))\overline{\overline{W_{\delta_{c}^{2}}^{+}}}where delta_(c)^(2)=(0,2c)\delta_{c}^{2}=(0,2 c). We can generalize this as follows.
Conjecture 3.19. Let c in Cc \in C and G\mathrm{G} a reductive group. Then we have the following conjectures:
(1) For any mu inX_(+)(G^(vv))\mu \in X_{+}\left(\mathrm{G}^{\vee}\right), the support of the mirror of rho_(mu)(E_(c))\rho_{\mu}\left(\mathbb{E}_{c}\right) is bar(bar(W_(delta_(c)^(mu))^(+)))\overline{\overline{W_{\delta_{c}^{\mu}}^{+}}}.
(2) Let mu inX_(+)(G^(vv))\mu \in X_{+}\left(\mathrm{G}^{\vee}\right) such that the corresponding irreducible G^(vv)\mathrm{G}^{\vee} representation rho_(mu)\rho_{\mu} is multiplicity free. Then the mirror of rho_(mu)(E_(c))\rho_{\mu}\left(\mathbb{E}_{c}\right) is O bar(bar(W_(delta_(c)^(+))^(+)))\mathcal{O} \overline{\overline{W_{\delta_{c}^{+}}^{+}}}.
(3) In the latter case, the multiplicity algebra of the restricted Hitchin map h_(G)h_{\mathrm{G}} : bar(bar(W_(delta_(c)^(mu))^(+)))rarrA\overline{\overline{W_{\delta_{c}^{\mu}}^{+}}} \rightarrow \mathcal{A} is isomorphic with the cohomology ring of bar(Gr)^(mu)\overline{\mathrm{Gr}}^{\mu}.
[1] V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. I. Monogr. Math. 82, Birkhäuser Boston, Inc., Boston, MA, 1985 .
[2] D. Baraglia and L. Schaposnik, Real structures on moduli spaces of Higgs bundles. Adv. Theor. Math. Phys. 20 (2016), no. 3, 525-551.
[3] V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3 (1994), no. 3, 493-535.
[4] V. Batyrev, Birational Calabi-Yau nn-folds have equal Betti numbers. In New trends in algebraic geometry (Warwick, 1996), pp. 1-11, London Math. Soc. Lecture Note Ser. 264, Cambridge Univ. Press, Cambridge, 1999.
[5] V. Batyrev and L. Borisov, Mirror duality and string-theoretic Hodge numbers. Invent. Math. 126 (1996), no. 1, 183-203.
[6] A. Beilinson and V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves. Preprint, 1995. https://math.uchicago.edu/ drinfeld/ langlands/QuantizationHitchin.pdf.
[7] A. Białynicki-Birula, Some theorems on actions of algebraic groups. Ann. of Math. (2) 98 (1973), 480-497.
[8] I. Biswas, O. GarcÃa-Prada, and J. Hurtubise, Higgs bundles, branes and Langlands duality. Comm. Math. Phys. 365 (2019), no. 3, 1005-1018.
[9] P. Candelas, M. Lynker, and R. Schimmrigk, Calabi-Yau manifolds in weighted P^(4)\mathbb{P}^{4}. Nuclear Phys. B 341 (1990), no. 2, 383-402.
[11] R. Donagi, Seiberg-Witten integrable systems. In Surveys in differential geometry: integral systems, pp. 83-129, Surv. Differ. Geom. 4, Int. Press, Boston, MA, 1998 .
[12] R. Donagi and E. Markman, Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles. In Integrable systems and quantum groups (Montecatini Terme, 1993), pp. 1-119, Lecture Notes in Math. 1620, Springer, Berlin, 1996.
[13] R. Donagi and T. Pantev, Langlands duality for Hitchin systems. Invent. Math. 189 (2006), no. 3, 653-735. arXiv:math/0604617.
[14] V. G. Drinfeld, Letter to P. Deligne, June 1981.
[15] V. G. Drinfeld, Two dimensional â„“\ell-adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2). Amer. J. Math. 105 (1983), 85-114.
[16] G. Faltings, Stable GG-bundles and projective connections. J. Algebraic Geom. 2 (1993), 507-568.
[17] B. Feigin, E. Frenkel, and L. Rybnikov, Opers with irregular singularity and spectra of the shift of argument subalgebra. Duke Math. J. 155 (2010), no. 2, 337-363.
[18] E. Frenkel, Lectures on the Langlands program and conformal field theory. In Frontiers in number theory, physics, and geometry. II, pp. 387-533, Springer, Berlin, 2007.
[19] S. Gelbart, An elementary introduction to the Langlands program. Bull. Amer. Math. Soc. (N.S.) 10, no. 2, 177-219.
[20] V. Ginzburg, Loop Grassmannian cohomology, the principal nilpotent and Kostant theorem. 1998, arXiv:math/9803141.
[21] U. Görtz, Affine Springer fibers and affine Deligne-Lusztig varieties. In Affine flag manifolds and principal bundles, pp. 1-50, Trends Math., Springer, Basel AG, Basel, 2010.
[22] B. R. Greene and M. R. Plesser, Duality in Calabi-Yau moduli space. Nuclear Phys. B 338 (1990), no. 1, 15-37.
[23] M. Gröchenig, D. Wyss, and P. Ziegler, Geometric stabilisation via pp-adic integration. J. Amer. Math. Soc. 33 (2020), no. 3, 807-873.
[24] M. Gröchenig, D. Wyss, and P. Ziegler, Mirror symmetry for moduli spaces of Higgs bundles via pp-adic integration. Invent. Math. 221 (2020), no. 2, 505-596.
[25] M. Gross, P. Hacking, S. Keel, and M. Kontsevich, Canonical bases for cluster algebras. J. Amer. Math. Soc. 31 (2018), no. 2, 497-608.
[26] M. Gross and B. Siebert, Affine manifolds, log structures, and mirror symmetry. Turkish J. Math. 27 (2003), no. 1, 33-60.
[27] M. Gross and B. Siebert, From real affine geometry to complex geometry. Ann. of Math. (2011), 1301-1428.
[28] T. Hausel, Global Topology of the Hitchin system, to appear in Handbook of moduli. In dedicated to David Mumford, edited by G. Farkas and I. Morrison, pp. 29-69, Adv. Lectures Math. 25, International Press, 2013, arXiv:1102.1717.
[29] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry. 2021, arXiv:2101.08583. To appear in Invent. Math.
[30] T. Hausel and N. Hitchin, Multiplicity algebras in integrable systems, ongoing project.
[31] T. Hausel, A. Mellit, and D. Pei, Mirror symmetry with branes by equivariant Verlinde formulas. In Geometry and physics. Vol. I, edited by J. E. Andersen et al., pp. 189-218, Oxford Univ. Press, Oxford, 2018.
[33] T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality and Hitchin systems. Invent. Math. 153 (2003), no. 1, 197-229.
[34] N. Hitchin, The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. (3) 55 (1987), 59-126.
[35] N. Hitchin, Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91-11491-114.
[36] N. Hitchin, Higgs bundles and characteristic classes. In Arbeitstagung Bonn 2013; In Memory of Friedrich Hirzebruch, pp. 247-264, Progr. Math., Birkhäuser, 2016, arXiv:1308.4603.
[37] N. Hitchin, Spinors, twistors and classical geometry. SIGMA Symmetry Integrability Geom. Methods Appl. 17 (2021), 090, 9 pp.
[38] K. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1 (2007), no. 1, 1-236.
[39] M. Kontsevich, Homological algebra of mirror symmetry. In Proceedings of the International Congress of Mathematicians (Zürich, 1994), pp. 120-139, Birkhäuser, Basel, 1995.
[41] G. Laumon, Un analogue du cone nilpotent. Duke Math. J. (2) 57 (1988), 647-671.
[42] D. Maulik and J. Shen, Endoscopic decompositions and the Hausel-Thaddeus conjecture. Forum Math. Pi 9 (2021), e8, 49 pp.
[43] C. Montonen and C. Olive, Magnetic monopoles as gauge particles? Phys. Lett. B 72 (1977), 117-120.
[44] S. Mukai, Duality between D(X)D(X) and D( hat(X))D(\hat{X}) with its application to Picard sheaves. Nagoya Math. J. 81 (1981), 153-175.
[45] B. C. Ngô, Le lemme fondamentale pour les algèbres de Lie. Publ. Math. Inst. Hautes Études Sci. 111 (2010), 1-169.
[46] N. Nitsure, Moduli space of semistable pairs on a curve. Proc. Lond. Math. Soc. (3) 62 (1991), 275-300.
[47] D. Panyushev, Weight multiplicity free representations, gg-endomorphism algebras, and Dynkin polynomials. J. Lond. Math. Soc. (2) 69 (2004), no. 2, 273-290.
[48] C. Simpson, Nonabelian Hodge theory, In Proceedings of the International Congress of Mathematicians (Kyoto, 1990), pp. 747-756, Math. Soc. Japan, 1991 .
[49] A. Strominger, S-T. Yau, and E. Zaslow, Mirror symmetry is T-duality. Nucl. Phys. B 479 (1996), 243-259.
[50] C. Yuen, Jet Schemes and Truncated Wedge Schemes. Ph.D. thesis, University of Michigan, 2006. http://www.math.lsa.umich.edu/ kesmith/YuenThesis.pdf.
[51] H. Zelaci, On very stablity of principal G-bundles. Geom. Dedicata 204 (2020), 165-173165-173.
[52] X. Zhu, Introduction to affine Grassmannians and the geometric Satake equivalence. In Geometry of moduli spaces and representation theory, pp. 59-154, IAS/Park City Math. Ser. 24, Amer. Math. Soc., Providence, RI, 2017.
TAMÃS HAUSEL
Institute of Science and Technology Austria, Am Campus 1, Klosterneuburg 3400, Austria, tamas.hausel@ist.ac.at
HODGE THEORY, BETWEEN ALGEBRAICITY AND TRANSCENDENCE
BRUNO KLINGLER
ABSTRACT
The Hodge theory of complex algebraic varieties is at heart a transcendental comparison of two algebraic structures. We survey the recent advances bounding this transcendence, mainly due to the introduction of o-minimal geometry as a natural framework for Hodge theory.
MATHEMATICS SUBJECT CLASSIFICATION 2020
Primary 14D; Secondary 14D07, 14C30, 32G20
KEYWORDS
Hodge theory, variations of Hodge structures, periods
1. INTRODUCTION
Let XX be a smooth connected projective variety over C\mathbb{C}, and X^("an ")X^{\text {an }} its associated compact complex manifold. Classical Hodge theory [52] states that the Betti (i.e., singular) cohomology group H_(B)^(k)(X^("an "),Z)H_{\mathrm{B}}^{k}\left(X^{\text {an }}, \mathbb{Z}\right) is a polarizable Z\mathbb{Z}-Hodge structure of weight kk : there exists a canonical decomposition (called the Hodge decomposition) of complex vector spaces
and a (-1)^(k)(-1)^{k}-symmetric bilinear pairing q_(k):H_(B)^(k)(X^("an "),Z)xxH_(B)^(k)(X^("an "),Z)rarrZq_{k}: H_{\mathrm{B}}^{k}\left(X^{\text {an }}, \mathbb{Z}\right) \times H_{\mathrm{B}}^{k}\left(X^{\text {an }}, \mathbb{Z}\right) \rightarrow \mathbb{Z} whose complexification makes the above decomposition orthogonal, and satisfies the positivity condition (the signs are complicated but are imposed to us by geometry)
i^(p-q)q_(k,C)(alpha, bar(alpha)) > 0quad" for any nonzero "alpha inH^(p,q)(X^(an))\mathrm{i}^{p-q} q_{k, \mathbb{C}}(\alpha, \bar{\alpha})>0 \quad \text { for any nonzero } \alpha \in H^{p, q}\left(X^{\mathrm{an}}\right)
Deligne [29] vastly generalized Hodge's result, showing that the cohomology H_(B)^(k)(X^("an "),Z)H_{\mathrm{B}}^{k}\left(X^{\text {an }}, \mathbb{Z}\right) of any complex algebraic variety XX is functorially endowed with a slightly more general graded polarizable mixed Z\mathbb{Z}-Hodge structure, that makes, after tensoring with Q,H_(B)^(k)(X^(an),Q)\mathbb{Q}, H_{\mathrm{B}}^{k}\left(X^{\mathrm{an}}, \mathbb{Q}\right) a successive extension of polarizable Q\mathbb{Q}-Hodge structures, with weights between 0 and 2k2 k. As mixed Q\mathbb{Q}-Hodge structures form a Tannakian category MHS_(Q)\mathrm{MHS}_{\mathbb{Q}}, one can conveniently (although rather abstractly) summarize the Hodge-Deligne theory as functorially assigning to any complex algebraic variety XX a Q\mathbb{Q}-algebraic group: the Mumford-Tate group MT_(X)\mathbf{M T}_{X} of XX, defined as the Tannaka group of the Tannakian subcategory (:H_(B)^(∙)(X^("an "),Q):)\left\langle H_{\mathrm{B}}^{\bullet}\left(X^{\text {an }}, \mathbb{Q}\right)\right\rangle of MHS_(Q)\mathrm{MHS}_{\mathbb{Q}} generated by H_(B)^(∙)(X^("an "),Q)H_{\mathrm{B}}^{\bullet}\left(X^{\text {an }}, \mathbb{Q}\right). The knowledge of the group MT_(X)\mathbf{M} \mathbf{T}_{X} is equivalent to the knowledge of all Hodge tensors for the Hodge structure H_(B)^(∙)(X^("an "),Q)H_{\mathrm{B}}^{\bullet}\left(X^{\text {an }}, \mathbb{Q}\right).
These apparently rather innocuous semilinear algebra statements are anything but trivial. They have become the main tool for analyzing the topology, geometry and arithmetic of complex algebraic varieties. Let us illustrate what we mean with regard to topology, which we will not go into later. The existence of the Hodge decomposition for smooth projective complex varieties, which holds more generally for compact Kähler manifolds, imposes many constraints on the cohomology of such spaces, the most obvious being that their odd Betti numbers have to be even. Such constraints are not satisfied even by compact complex manifolds as simple as the Hopf surfaces, quotients of C^(2)\\{0}\mathbb{C}^{2} \backslash\{0\} by the action of Z\mathbb{Z} given by multiplication by lambda!=0,|lambda|!=1\lambda \neq 0,|\lambda| \neq 1, whose first Betti number is one. Characterizing the homotopy types of compact Kähler manifolds is an essentially open question, which we will not discuss here.
The mystery of the Hodge-Deligne theory lies in the fact that it is at heart not an algebraic theory, but rather the transcendental comparison of two algebraic structures. For simplicity, let XX be a smooth connected projective variety over C\mathbb{C}. The Betti cohomology H_(B)^(∙)(X^("an "),Q)H_{\mathrm{B}}^{\bullet}\left(X^{\text {an }}, \mathbb{Q}\right) defines a Q\mathbb{Q}-structure on the complex vector space of the algebraic de Rham cohomology H_(dR)^(∙)(X//C):=H^(∙)(X,Omega_(X//C)^(∙))H_{\mathrm{dR}}^{\bullet}(X / \mathbb{C}):=H^{\bullet}\left(X, \Omega_{X / \mathbb{C}}^{\bullet}\right) under the transcendental comparison isomorphism:
where the first canonical isomorphism is the comparison between algebraic and analytic de Rham cohomology provided by GAGA, and the second one is provided by integrating complex C^(oo)\mathrm{C}^{\infty} differential forms over cycles (de Rham's theorem). The Hodge filtration F^(p)F^{p} on H_(B)^(∙)(X^("an "),Q)ox_(Q)CH_{\mathrm{B}}^{\bullet}\left(X^{\text {an }}, \mathbb{Q}\right) \otimes_{\mathbb{Q}} \mathbb{C} is the image under ϖ\varpi of the algebraic filtration F^(p)=F^{p}=Im(H^(∙)(X,Omega_(X//C)^(∙ >= p))rarrH_(dR)^(∙)(X//C))\operatorname{Im}\left(H^{\bullet}\left(X, \Omega_{X / \mathbb{C}}^{\bullet \geq p}\right) \rightarrow H_{\mathrm{dR}}^{\bullet}(X / \mathbb{C})\right) on the left-hand side.
The surprising power of the Hodge-Deligne theory lies in the fact that, although the comparison between the two algebraic structures is transcendental, this transcendence should be severely constrained, as predicted, for instance, by the Hodge conjecture and the Grothendieck period conjecture:
For XX smooth projective, it is well known that the cycle class [Z][Z] of any codimension kk algebraic cycle on XX with Q\mathbb{Q} coefficients is a Hodge class in the Hodge structure H^(2k)(X^("an "),Q)(k)H^{2 k}\left(X^{\text {an }}, \mathbb{Q}\right)(k). Hodge [52] famously conjectured that the converse holds true: any Hodge class in H^(2k)(X,Q)(k)H^{2 k}(X, \mathbb{Q})(k) should be such a cycle class.
For XX smooth and defined over a number field K subCK \subset \mathbb{C}, its periods are the coefficients of the matrix of Grothendieck's isomorphism (generalizing (1.1))
with respect to bases of H_(dR)^(∙)(X//K)H_{\mathrm{dR}}^{\bullet}(X / K) and H_(B)^(∙)(X^("an "),Q)H_{\mathrm{B}}^{\bullet}\left(X^{\text {an }}, \mathbb{Q}\right). The Grothendieck period conjecture (combined with the Hodge conjecture) predicts that the transcendence degree of the field k_(X)subCk_{X} \subset \mathbb{C} generated by the periods of XX coincides with the dimension of MT_(X)\mathbf{M} \mathbf{T}_{X}.
This tension between algebraicity and transcendence is perhaps best revealed when considering Hodge theory in families, as developed by Griffiths [43]. Let f:X rarr Sf: X \rightarrow S be a smooth projective morphism of smooth connected quasiprojective varieties over C\mathbb{C}. Its complex analytic fibers X_(s)^(an),s inS^(an)X_{s}^{\mathrm{an}}, s \in S^{\mathrm{an}}, are diffeomorphic, hence their cohomologies V_(Z,s):=H_(B)^(∙)(X_(s)^(an),Z)\mathbb{V}_{\mathbb{Z}, s}:=H_{\mathrm{B}}^{\bullet}\left(X_{s}^{\mathrm{an}}, \mathbb{Z}\right), s inS^("an ")s \in S^{\text {an }} are all isomorphic to a fixed abelian group V_(Z)V_{\mathbb{Z}} and glue together into a locally constant sheaf V_(Z):=R^(∙)f^("an ")_(**)\mathbb{V}_{\mathbb{Z}}:=R^{\bullet} f^{\text {an }}{ }_{*} on S^("an ")S^{\text {an }}. However, the complex algebraic structure on X_(s)X_{s}, hence also the Hodge structure on V_(Z,s)\mathbb{V}_{\mathbb{Z}, s}, varies with ss, making R^(∙)f_(**)^("an ")ZR^{\bullet} f_{*}^{\text {an }} \mathbb{Z} a variation of Z\mathbb{Z}-Hodge structures ( ZVHS)V\mathbb{Z V H S}) \mathbb{V} on S^("an ")S^{\text {an }}, which can be naturally polarized. One easily checks that the Mumford-Tate group G_(s):=MT_(X_(s)),s inS^("an ")\mathbf{G}_{s}:=\mathbf{M T}_{X_{s}}, s \in S^{\text {an }}, is locally constant equal to the so-called generic Mumford-Tate group G\mathbf{G}, outside of a meagre set HL(S,f)subS^("an ")\operatorname{HL}(S, f) \subset S^{\text {an }}, the Hodge locus of the morphism ff, where it shrinks as exceptional Hodge tensors appear in H_(B)^(∙)(X_(s)^(an),Z)H_{\mathrm{B}}^{\bullet}\left(X_{s}^{\mathrm{an}}, \mathbb{Z}\right). The variation V\mathbb{V} is completely described by its period map
Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash D
Here the period domain DD classifies all possible Z\mathbb{Z}-Hodge structure on the abelian group V_(Z)V_{\mathbb{Z}}, with a fixed polarization and Mumford-Tate group contained in G\mathbf{G}; and Phi\Phi maps a point s inS^("an ")s \in S^{\text {an }} to the point of DD parameterizing the polarized Z\mathbb{Z}-Hodge structure on V_(Z)V_{\mathbb{Z}} defined by V_(Z,s)\mathbb{V}_{\mathbb{Z}, s} (well defined up to the action of the arithmetic group Gamma:=G nnGL(V_(Z))\Gamma:=G \cap \mathbf{G L}\left(V_{\mathbb{Z}}\right) ).
The transcendence of the comparison isomorphism (1.1) for each fiber X_(s)X_{s} is embodied in the fact that the Hodge variety Gamma\\D\Gamma \backslash D is, in general, a mere complex analytic variety
not admitting any algebraic structure; and that the period map Phi\Phi is a mere complex analytic map. On the other hand this transcendence is sufficiently constrained so that the following corollary of the Hodge conjecture [96] holds true, as proven by Cattani-Deligne-Kaplan [22]: the Hodge locus HL(S,f)\mathrm{HL}(S, f) is a countable union of algebraic subvarieties of SS. Remarkably, their result is in fact valid for any polarized ZVHSV\mathbb{Z} \mathrm{VHS} \mathbb{V} on S^("an ")S^{\text {an }}, not necessarily coming from geometry: the Hodge locus HL(S,V^(ox))\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right) is a countable union of algebraic subvarieties of SS.
In this paper we report on recent advances in the understanding of this interplay between algebraicity and transcendence in Hodge theory, our main object of interest being period maps Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash D. The paper is written for nonexperts: we present the mathematical objects involved, the questions, and the results but give only vague ideas of proofs, if any. It is organized as follows. After Section 2 presenting the objects of Hodge theory (which the advanced reader will skip to refer to on occasion), we present in Section 3 the main driving force behind the recent advances: although period maps are very rarely complex algebraic, their geometry is tame and does not suffer from any of the many possible pathologies of a general holomorphic map. In model-theoretic terms, period maps are definable in the o-minimal structure R_(an,exp)\mathbb{R}_{\mathrm{an}, \mathrm{exp}}. In Section 4 , we introduce the general format of bialgebraic structures for comparing the algebraic structure on SS and that on (the compact dual D^(ˇ)\check{D} of) the period domain DD. The heuristic provided by this format, combined with o-minimal geometry, leads to a powerful functional transcendence result: the Ax-Schanuel theorem for polarized ZVHS\mathbb{Z V H S}. It also suggests to interpret variational Hodge theory as a special case of an atypical intersection problem. In Section 5 we describe how this viewpoint leads to a stunning improvement of the result of Cattani, Deligne, and Kaplan: in most cases HL(S,V^(ox))\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right) is not only a countable union of algebraic varieties, but is actually algebraic on the nose (at least if we restrict to its components of positive period dimension). Finally, in Section 6 we turn briefly to some arithmetic aspects of the theory.
For the sake of simplicity, we focus on the case of pure Hodge structures, only mentioning the references dealing with the mixed case.
2. VARIATIONS OF HODGE STRUCTURES AND PERIOD MAPS
2.1. Polarizable Hodge structures
Let n inZn \in \mathbb{Z}. Let R=Z,QR=\mathbb{Z}, \mathbb{Q}, or R\mathbb{R}. An RR-Hodge structure VV of weight nn is a finitely generated RR-module V_(R)V_{R} together with one of the following equivalent data: a bigrading V_(C)(:=V_{\mathbb{C}}(:={:V_(R)ox_(R)C)=bigoplus_(p+q=n)V^(p,q)\left.V_{R} \otimes_{R} \mathbb{C}\right)=\bigoplus_{p+q=n} V^{p, q}, called the Hodge decomposition, such that bar(V^(p,q))=V^(q,p)\overline{V^{p, q}}=V^{q, p} (the numbers (dim V^(p,q))_(p+q=n)\left(\operatorname{dim} V^{p, q}\right)_{p+q=n} are called the Hodge numbers of {:V)\left.V\right); or a decreasing filtration F^(∙)F^{\bullet} of V_(C)V_{\mathbb{C}}, called the Hodge filtration, satisfying F^(p)o+ bar(F^(n+1-p))=V_(C)F^{p} \oplus \overline{F^{n+1-p}}=V_{\mathbb{C}}. One goes from one to the other through F^(p)=bigoplus_(r >= p)V^(r,n-r)F^{p}=\bigoplus_{r \geq p} V^{r, n-r} and V^(p,q)=F^(p)nn bar(F^(q))V^{p, q}=F^{p} \cap \overline{F^{q}}. The following group-theoretic description will be most useful to us: a Hodge structure is an RR-module V_(R)V_{R} and a real algebraic representation varphi:SrarrGL(V_(R))\varphi: \mathbf{S} \rightarrow \mathbf{G L}\left(V_{\mathbb{R}}\right) whose restriction to G_(m,R)\mathbf{G}_{m, \mathbb{R}} is defined over Q\mathbb{Q}. Here the Deligne torus S\mathbf{S} denotes the real algebraic group C^(**)\mathbb{C}^{*} of invertible matrices of the forms ([a,-b],[b,a])\left(\begin{array}{cc}a & -b \\ b & a\end{array}\right), which contains the diagonal subgroup G_(m,R)\mathbf{G}_{m, \mathbb{R}}. Being of weight nn is the requirement
that varphi_(∣G_(m,R))\varphi_{\mid \mathbf{G}_{m, \mathbb{R}}} acts via the character z|->z^(-n)z \mapsto z^{-n}. The space V^(p,q)V^{p, q} is recovered as the eigenspace for the character z|->z^(-p_(z)) bar(z)^(-q)z \mapsto z^{-p_{z}} \bar{z}^{-q} of S(R)≃C^(**)\mathbf{S}(\mathbb{R}) \simeq \mathbb{C}^{*}. A morphism of Hodge structures is a morphism of RR-modules compatible with the bigrading (equivalently, with the Hodge filtration or the S-action).
Example 2.1. We write R(n)R(n) for the unique RR-Hodge structure of weight -2n-2 n, called the Tate-Hodge structure of weight -2n-2 n, on the rank-one free RR-module (2pii)^(n)R subC(2 \pi \mathrm{i})^{n} R \subset \mathbb{C}.
One easily checks that the category of RR-Hodge structures is an abelian category (where the kernels and cokernels coincide with the usual kernels and cokernels in the category of RR-modules, with the induced Hodge filtrations on their complexifications), with natural tensor products V ox WV \otimes W and internal homs hom (V,W)(V, W) (in particular, duals V^(vv):=V^{\vee}:= hom (V,R(0)))(V, R(0))). For R=QR=\mathbb{Q}, or R\mathbb{R}, we obtain a Tannakian category, with an obvious exact faithful RR-linear tensor functor omega:(V_(R),varphi)|->V_(R)\omega:\left(V_{R}, \varphi\right) \mapsto V_{R}. In particular, R(n)=R(1)^(ox n)R(n)=R(1)^{\otimes n}. If VV is an RR-Hodge structure, we write V(n):=V ox R(n)V(n):=V \otimes R(n) its nnth Tate twist.
If V=(V_(R),varphi)V=\left(V_{R}, \varphi\right) is an RR-Hodge structure of weight nn, a polarization for VV is a morphism of RR-Hodge structures q:V^(ox2)rarr R(-n)q: V^{\otimes 2} \rightarrow R(-n) such that (2pii)^(n)q(x,varphi(i)y)(2 \pi \mathrm{i})^{n} q(x, \varphi(\mathrm{i}) y) is a positivedefinite bilinear form on V_(R)V_{\mathbb{R}}, called the Hodge form associated with the polarization. If there exists a polarization for VV then VV is said polarizable. One easily checks that the category of polarizable Q\mathbb{Q}-Hodge structures is semisimple.
Example 2.2. Let MM be a compact complex manifold. If MM admits a Kähler metric, the singular cohomology H_(B)^(n)(M,Z)H_{\mathrm{B}}^{n}(M, \mathbb{Z}) is naturally a Z\mathbb{Z}-Hodge structure of weight nn, see [52], [94, CHAP. 6]:
where H_(dR)^(∙)(M,C)H_{d R}^{\bullet}(M, \mathbb{C}) denotes the de Rham cohomology of the complex (A^(∙)(M,C),d)\left(A^{\bullet}(M, \mathbb{C}), d\right) of C^(oo)\mathrm{C}^{\infty} differential forms on MM, the first equality is the canonical isomorphism obtained by integrating forms on cycles (de Rham theorem), and the complex vector subspace H^(p,q)(M)H^{p, q}(M) of H_(dR)^(n)(M,C)H_{\mathrm{dR}}^{n}(M, \mathbb{C}) is generated by the dd-closed forms of type (p,q)(p, q), and thus satisfies automatically bar(H^(p,q)(M))=H^(q,p)(M)\overline{H^{p, q}(M)}=H^{q, p}(M). Although the second equality depends only on the complex structure on MM, its proof relies on the choice of a Kähler form omega\omega on MM through the following sequence of isomorphisms:
where H_(Delta_(omega))^(n)(M)\mathscr{H}_{\Delta_{\omega}}^{n}(M) denotes the vector space of Delta_(omega)\Delta_{\omega}-harmonic differential forms on MM and H_(Delta_(omega))^(p,q)(M)\mathscr{H}_{\Delta_{\omega}}^{p, q}(M) its subspace of Delta_(omega)\Delta_{\omega}-harmonic (p,q)(p, q)-forms. The heart of Hodge theory is thus reduced to the statement that the Laplacian Delta_(omega)\Delta_{\omega} of a Kähler metric preserves the type of forms. The choice of a Kähler form omega\omega on MM also defines, through the hard Lefschetz theorem [94, THEOREM 6.25], a polarization of the R\mathbb{R}-Hodge structure H^(n)(M,R)H^{n}(M, \mathbb{R}), see [94, THEOREM 6.32]. If f:M rarr Nf: M \rightarrow N is any holomorphic map between compact complex manifolds admitting Kähler metrics then both f^(**):H_(B)^(n)(N,Z)rarrH_(B)^(n)(M,Z)f^{*}: H_{\mathrm{B}}^{n}(N, \mathbb{Z}) \rightarrow H_{\mathrm{B}}^{n}(M, \mathbb{Z}) and the Gysin morphism f_(**):H_(B)^(n)(M,Z)rarrf_{*}: H_{\mathrm{B}}^{n}(M, \mathbb{Z}) \rightarrowH_(B)^(n-2r)(N,Z)(-r)H_{\mathrm{B}}^{n-2 r}(N, \mathbb{Z})(-r) are morphism of Z\mathbb{Z}-Hodge structures, where r=dim M-dim Nr=\operatorname{dim} M-\operatorname{dim} N.
Example 2.3. Suppose moreover that M=X^("an ")M=X^{\text {an }} is the compact complex manifold analytification of a smooth projective variety XX over C\mathbb{C}. In that case, H_(B)^(n)(X,Z)H_{\mathrm{B}}^{n}(X, \mathbb{Z}) is a polarizable Z\mathbb{Z}-Hodge structure. Indeed, the Kähler class [omega][\omega] can be chosen as the first Chern class of an ample line bundle on XX, giving rise to a rational Lefschetz decomposition and (after clearing denominators by multiplying by a sufficiently large integer) to an integral polarization. Moreover, the Hodge filtration F^(∙)F^{\bullet} on H_(B)^(n)(X^("an "),C)H_{\mathrm{B}}^{n}\left(X^{\text {an }}, \mathbb{C}\right) can be defined algebraically: upon identifying H_(B)^(n)(X^("an "),C)H_{\mathrm{B}}^{n}\left(X^{\text {an }}, \mathbb{C}\right) with the algebraic de Rham cohomology H_(dR)^(n)(X//C):=H^(n)(X,Omega_(X//C)^(∙))H_{\mathrm{dR}}^{n}(X / \mathbb{C}):=H^{n}\left(X, \Omega_{X / \mathbb{C}}^{\bullet}\right), the Hodge filtration is given by F^(p)=Im(H^(n)(X,Omega_(X//C)^(∙ >= p))rarrH_(B)^(n)(X^("an "),C))F^{p}=\operatorname{Im}\left(H^{n}\left(X, \Omega_{X / \mathbb{C}}^{\bullet \geq p}\right) \rightarrow H_{\mathrm{B}}^{n}\left(X^{\text {an }}, \mathbb{C}\right)\right). It follows that if XX is defined over a subfield KK of C\mathbb{C}, then the Hodge filtration F^(∙)F^{\bullet} on H_(B)^(n)(X^("an "),C)=H_{\mathrm{B}}^{n}\left(X^{\text {an }}, \mathbb{C}\right)=H_(dR)^(n)(X//K)ox_(K)CH_{\mathrm{dR}}^{n}(X / K) \otimes_{K} \mathbb{C} is defined over KK.
Example 2.4. The functor which assigns to a complex abelian variety AA its H_(B)^(1)(A^("an "),Z)H_{\mathrm{B}}^{1}\left(A^{\text {an }}, \mathbb{Z}\right) defines an equivalence of categories between abelian varieties and polarizable Z\mathbb{Z}-Hodge structures of weight 1 and type (1,0)(1,0) and (0,1)(0,1).
2.2. Hodge classes and Mumford-Tate group
Let R=ZR=\mathbb{Z} or Q\mathbb{Q} and let VV be an RR-Hodge structure. A Hodge class for VV is a vector in V^(0,0)nnV_(Q)=F^(0)V_(C)nnV_(Q)V^{0,0} \cap V_{\mathbb{Q}}=F^{0} V_{\mathbb{C}} \cap V_{\mathbb{Q}}. For instance, any morphism of RR-Hodge structures f:V rarr Wf: V \rightarrow W defines a Hodge class in the internal hom(V,W)\operatorname{hom}(V, W). Let T^(m,n)V_(Q)T^{m, n} V_{\mathbb{Q}} denote the Q\mathbb{Q} Hodge structure V_(Q)^(ox m)ox hom(V,R(0))_(Q)^(ox n)V_{\mathbb{Q}}^{\otimes m} \otimes \operatorname{hom}(V, R(0))_{\mathbb{Q}}^{\otimes n}. A Hodge tensor for VV is a Hodge class in some T^(m,n)V_(Q)T^{m, n} V_{\mathbb{Q}}.
The main invariant of an RR-Hodge structure is its Mumford-Tate group. For any RR Hodge structure VV we denote by (:V:)\langle V\rangle the Tannakian subcategory of the category of Q\mathbb{Q}-Hodge structures generated by V_(Q)V_{\mathbb{Q}}; in other words, (:V:)\langle V\rangle is the smallest full subcategory containing V,Q(0)V, \mathbb{Q}(0) and stable under o+,ox\oplus, \otimes, and taking subquotients. If omega_(V)\omega_{V} denotes the restriction of the tensor functor omega\omega to (:V:)\langle V\rangle, the functor Aut^(ox)(omega_(V))\operatorname{Aut}^{\otimes}\left(\omega_{V}\right) is representable by some closed Q\mathbb{Q}-algebraic subgroup G_(V)subGL(V_(Q))\mathbf{G}_{V} \subset \mathbf{G L}\left(V_{\mathbb{Q}}\right), called the Mumford-Tate group of VV, and omega_(V)\omega_{V} defines an equivalence of categories (:V:)≃Rep_(Q)G_(V)\langle V\rangle \simeq \operatorname{Rep}_{\mathbb{Q}} \mathbf{G}_{V}. See [33, II, 2.11].
The Mumford-Tate group G_(V)\mathbf{G}_{V} can also be characterized as the fixator in GL(V_(Q))\mathbf{G L}\left(V_{\mathbb{Q}}\right) of the Hodge tensors for VV, or equivalently, writing V=(V_(R),varphi)V=\left(V_{R}, \varphi\right), as the smallest Q\mathbb{Q}-algebraic subgroup of GL(V_(Q))\mathbf{G L}\left(V_{\mathbb{Q}}\right) whose base change to R\mathbb{R} contains the image Im varphi\operatorname{Im} \varphi. In particular varphi\varphi factorizes as varphi:SrarrG_(V,R)\varphi: \mathbf{S} \rightarrow \mathbf{G}_{V, \mathbb{R}}. The group G_(V)\mathbf{G}_{V} is thus connected, and reductive if VV is polarizable. See [2, LEMMA 2].
Example 2.5. G_(Z(n))=G_(m)\mathbf{G}_{\mathbb{Z}(n)}=\mathbf{G}_{m} if n!=0n \neq 0 and G_(Z(0))={1}\mathbf{G}_{\mathbb{Z}(0)}=\{1\}.
Example 2.6. Let AA be a complex abelian variety and let V:=H_(B)^(1)(A^("an "),Z)V:=H_{\mathrm{B}}^{1}\left(A^{\text {an }}, \mathbb{Z}\right) be the associated Z\mathbb{Z}-Hodge structure of weight 1 . We write G_(A):=G_(V)\mathbf{G}_{A}:=\mathbf{G}_{V}. The choice of an ample line bundle on AA defines a polarization qq on VV. On the one hand, the endomorphism algebra D:=D:=End^(0)(A)(:=End(A)ox_(Z)Q)\operatorname{End}^{0}(A)\left(:=\operatorname{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}\right) is a finite-dimensional semisimple Q\mathbb{Q}-algebra which, in view of Example 2.4, identifies with End (V_(Q))^(G_(A))\operatorname{End}\left(V_{\mathbb{Q}}\right)^{\mathbf{G}_{A}}. Thus G_(A)subGL_(D)(V_(Q))\mathbf{G}_{A} \subset \mathbf{G L}_{D}\left(V_{\mathbb{Q}}\right). On the other hand, the polarization qq defines a Hodge class in hom(V_(Q)^(ox2),Q(-1))\operatorname{hom}\left(V_{\mathbb{Q}}^{\otimes 2}, \mathbb{Q}(-1)\right) thus G_(A)\mathbf{G}_{A} has to be contained in
the group GSp(V_(Q),q)\mathbf{G S p}\left(V_{\mathbb{Q}}, q\right) of symplectic similitudes of V_(Q)V_{\mathbb{Q}} with respect to the symplectic form qq. Finally, G_(A)subGL_(D)(V_(Q))nnGSp(V_(Q),q)\mathbf{G}_{A} \subset \mathbf{G L}_{D}\left(V_{\mathbb{Q}}\right) \cap \mathbf{G S p}\left(V_{\mathbb{Q}}, q\right).
If A=EA=E is an elliptic curve, it follows readily that either D=QD=\mathbb{Q} and G_(E)=GL_(2)\mathbf{G}_{E}=\mathbf{G L}_{2}, or DD is an imaginary quadratic field ( EE has complex multiplication) and G_(E)=T_(D)\mathbf{G}_{E}=\mathbf{T}_{D}, the Q\mathbb{Q}-torus defined by T_(D)(S)=(D oxQS)^(**)\mathbf{T}_{D}(S)=(D \otimes \mathbb{Q} S)^{*} for any Q\mathbb{Q}-algebra SS.
2.3. Period domains and Hodge data
Let V_(Z)V_{\mathbb{Z}} be a finitely generated abelian group V_(Z)V_{\mathbb{Z}} of rank rr. Fix a positive integer nn, a (-1)^(n)(-1)^{n}-symmetric bilinear form q_(Z)q_{\mathbf{Z}} on V_(Z)V_{\mathbb{Z}} and a collection of nonnegative integers (h^(p,q))(p,q >= 0,p+q=n)\left(h^{p, q}\right)(p, q \geq 0, p+q=n) such that h^(p,q)=h^(q,p)h^{p, q}=h^{q, p} and sumh^(p,q)=r\sum h^{p, q}=r. Associated with (n,q_(Z),(h^(p,q)):}\left(n, q_{\mathbb{Z}},\left(h^{p, q}\right)\right. ) we want to define a period domain DD classifying Z\mathbb{Z}-Hodge structures of weight nn on V_(Z)V_{\mathbb{Z}}, polarized by q_(Z)q_{\mathbb{Z}}, and with Hodge numbers h^(p,q)h^{p, q}. Setting f^(p)=sum_(r >= p)h^(r,n-r)f^{p}=\sum_{r \geq p} h^{r, n-r}, we first define the compact dual D^(ˇ)\check{D} parametrizing the finite decreasing filtrations F^(∙)F^{\bullet} on V_(C)V_{\mathbb{C}} satisfying (F^(p))^(_|__(q_(Z)))=F^(n+1-p)\left(F^{p}\right)^{\perp_{q_{\mathbb{Z}}}}=F^{n+1-p} and dim F^(p)=f^(p)\operatorname{dim} F^{p}=f^{p}. This is a closed algebraic subvariety of the product of Grassmannians prod_(p)Gr(f^(p),V_(C))\prod_{p} \operatorname{Gr}\left(f^{p}, V_{\mathbb{C}}\right). The period domain D subD^(ˇ)^("an ")D \subset \check{D}^{\text {an }} is the open subset where the Hodge form is positive definite. If G:=GAut(V_(Q),q_(Q))\mathbf{G}:=\operatorname{GAut}\left(V_{\mathbb{Q}}, q_{\mathbb{Q}}\right) denotes the group of similitudes of q_(Q)q_{\mathbb{Q}}, one easily checks that G(C)\mathbf{G}(\mathbb{C}) acts transitively on D^(ˇ)^("an ")\check{D}^{\text {an }}, which is thus a flag variety for G_(C)\mathbf{G}_{\mathbb{C}}; and that the connected component G:=G^("der ")(R)^(+)G:=\mathbf{G}^{\text {der }}(\mathbb{R})^{+}of the identity in the derived group G^("der ")(R)\mathbf{G}^{\text {der }}(\mathbb{R}) acts transitively on DD, which identifies with an open GG-orbit in D^(ˇ)\check{D}. If we fix a base point o in Do \in D and denote by PP and MM its stabilizer in G(C)\mathbf{G}(\mathbb{C}) and GG, respectively, the period domain DD is thus the homogeneous space
D=G//M↪D^(ˇ)^("an ")=G(C)//P". "D=G / M \hookrightarrow \check{D}^{\text {an }}=\mathbf{G}(\mathbb{C}) / P \text {. }
The group PP is a parabolic subgroup of G(C)\mathbf{G}(\mathbb{C}). Its subgroup M=P nn GM=P \cap G, consisting of real elements, not only fixes the filtration F_(o)^(∙)F_{o}^{\bullet} but also the Hodge decomposition, hence the Hodge form, at oo. It is thus a compact subgroup of GG and DD is an open elliptic orbit of GG in D^(ˇ)\check{D}.
More generally, let G\mathbf{G} be a connected reductive Q\mathbb{Q}-algebraic group and let varphi:Srarr\varphi: \mathbf{S} \rightarrowG_(R)\mathbf{G}_{\mathbb{R}} be a real algebraic morphism such that varphi_(∣G_(m,R))\varphi_{\mid \mathbf{G}_{m, \mathbb{R}}} is defined over Q\mathbb{Q}. We assume that G\mathbf{G} is the Mumford-Tate group of varphi\varphi. The period domain (or Hodge domain) DD associated with varphi:SrarrG_(R)\varphi: \mathbf{S} \rightarrow \mathbf{G}_{\mathbb{R}} is the connected component of the G(R)\mathbf{G}(\mathbb{R})-conjugacy class of varphi:SrarrG_(R)\varphi: \mathbf{S} \rightarrow \mathbf{G}_{\mathbb{R}} in Hom(S,G_(R))\operatorname{Hom}\left(\mathbf{S}, \mathbf{G}_{\mathbb{R}}\right). Again, one easily checks that DD is an open elliptic orbit of G:=G^(der)(R)^(+)G:=\mathbf{G}^{\mathrm{der}}(\mathbb{R})^{+} in the compact dual flag variety D^(ˇ)^("an ")\check{D}^{\text {an }}, the G(C)\mathbf{G}(\mathbb{C})-conjugacy class of varphi_(C)@mu:G_(m,C)rarrG_(C)\varphi_{\mathbb{C}} \circ \mu: \mathbf{G}_{\mathrm{m}, \mathbb{C}} \rightarrow \mathbf{G}_{\mathbb{C}}, where mu:G_(m,C)rarrS_(C)=G_(m,C)xxG_(m,C)\mu: \mathbf{G}_{\mathbf{m}, \mathbb{C}} \rightarrow \mathbf{S}_{\mathbb{C}}=\mathbf{G}_{\mathbf{m}, \mathbb{C}} \times \mathbf{G}_{\mathbf{m}, \mathbb{C}} is the cocharacter z|->(z,1)z \mapsto(z, 1). See [41] for details. The pair (G,D)(\mathbf{G}, D) is called a (connected) Hodge datum. A morphism of Hodge data (G,D)rarr(\mathbf{G}, D) \rightarrow (G^('),D^('))\left(\mathbf{G}^{\prime}, D^{\prime}\right) is a morphism rho:GrarrG^(')\rho: \mathbf{G} \rightarrow \mathbf{G}^{\prime} sending DD to D^(')D^{\prime}. Any linear representation lambda:Grarr\lambda: \mathbf{G} \rightarrowGL(V_(Q))\mathbf{G L}\left(V_{\mathbb{Q}}\right) defines a G(Q)\mathbf{G}(\mathbb{Q})-equivariant local system V^(ˇ)_(lambda)\check{\mathbb{V}}_{\lambda} on D^(ˇ)^("an ")\check{D}^{\text {an }}. Moreover, each point x inx \inDD, seen as a morphism varphi_(x):SrarrG_(R)\varphi_{x}: \mathbf{S} \rightarrow \mathbf{G}_{\mathbb{R}}, defines a Q\mathbb{Q}-Hodge structure V_(x):=(V_(Q),lambda@varphi_(x))V_{x}:=\left(V_{\mathbb{Q}}, \lambda \circ \varphi_{x}\right). The G(C)\mathbf{G}(\mathbb{C})-equivariant filtration F^(∙)V^(ˇ)_(lambda):=G^("ad ")(C)xx_(P,lambda)F^(∙)V_(o,C)F^{\bullet} \check{\mathcal{V}}_{\lambda}:=\mathbf{G}^{\text {ad }}(\mathbb{C}) \times_{P, \lambda} F^{\bullet} V_{o, \mathbb{C}} of the holomorphic vector bundle V^(ˇ)_(lambda):=G^("ad ")(C)xx_(P,lambda)V_(o,C)\check{\mathcal{V}}_{\lambda}:=\mathbf{G}^{\text {ad }}(\mathbb{C}) \times_{P, \lambda} V_{o, \mathbb{C}} on D^(ˇ)^("an ")\check{D}^{\text {an }} induces the Hodge filtration on V_(x)V_{x} for each x in Dx \in D. The Mumford-Tate group of V_(x)V_{x} is G\mathbf{G} precisely when x in D\\uuu tau(D^('))x \in D \backslash \bigcup \tau\left(D^{\prime}\right), where tau\tau ranges through the countable set of morphisms of Hodge data tau:(G^('),D^('))^(tau)rarr(G,D)\tau:\left(\mathbf{G}^{\prime}, D^{\prime}\right)^{\tau} \rightarrow(\mathbf{G}, D). The complex analytic subvarieties tau(D^('))\tau\left(D^{\prime}\right) of DD are called the special subvarieties of DD.
The following geometric feature of D^(ˇ)\check{D} will be crucial for us. The algebraic tangent bundle TD^(ˇ)T \check{D} naturally identifies, as a G_(C)\mathbf{G}_{\mathbb{C}}-equivariant bundle, with the quotient vector bundle V^(ˇ)_("Ad ")//F^(0)V^(ˇ)_("Ad ")\check{\mathcal{V}}_{\text {Ad }} / F^{0} \check{\mathcal{V}}_{\text {Ad }}, where Ad:GrarrGL(g)\mathrm{Ad}: \mathbf{G} \rightarrow \mathbf{G L}(\mathrm{g}) is the adjoint representation on the Lie algebra g\mathrm{g} of G\mathbf{G} In particular, it is naturally filtered by the F^(i)TD^(ˇ):=F^(i)V^(ˇ)_(Ad)//F^(0)V^(ˇ)_(Ad),i <= -1F^{i} T \check{D}:=F^{i} \check{\mathcal{V}}_{\mathrm{Ad}} / F^{0} \check{\mathcal{V}}_{\mathrm{Ad}}, i \leq-1. The subbundle F^(-1)TD^(ˇ)F^{-1} T \check{D} is called the horizontal tangent bundle of D^(ˇ)\check{D}.
2.4. Hodge varieties
Let (G,D)(\mathbf{G}, D) be a Hodge datum as in Section 2.3. A Hodge variety is the quotient Gamma\\D\Gamma \backslash D of DD by an arithmetic lattice Gamma\Gamma of G(Q)^(+):=G(Q)nn G\mathbf{G}(\mathbb{Q})^{+}:=\mathbf{G}(\mathbb{Q}) \cap G. It is thus naturally a complex analytic variety, which is smooth if Gamma\Gamma is torsion-free. The special subvarieties of Gamma\\D\Gamma \backslash D are the images of the special subvarieties of DD under the projection pi:D rarr Gamma\\D\pi: D \rightarrow \Gamma \backslash D (one easily checks these are closed complex analytic subvarieties of Gamma\\D)\Gamma \backslash D). For any algebraic representation lambda:GrarrGL(V_(Q))\lambda: \mathbf{G} \rightarrow \mathbf{G L}\left(V_{\mathbb{Q}}\right), the G(Q)\mathbf{G}(\mathbb{Q})-equivariant local system widetilde(V)_(lambda)\widetilde{V}_{\lambda} as well as the filtered holomorphic vector bundle (V^(ˇ)_(lambda),F^(∙))\left(\check{\mathcal{V}}_{\lambda}, F^{\bullet}\right) on D^(ˇ)\check{D} are GG-equivariant when restricted to DD, hence descend to a triple (V_(lambda),(V_(lambda),F^(∙)),grad)\left(\mathbb{V}_{\lambda},\left(\mathcal{V}_{\lambda}, F^{\bullet}\right), \nabla\right) on Gamma\\D\Gamma \backslash D. Similarly, the horizontal tangent bundle of D^(ˇ)\check{D} defines the horizontal tangent bundle T_(h)(Gamma\\D)sub T(Gamma\\D)T_{h}(\Gamma \backslash D) \subset T(\Gamma \backslash D) of the Hodge variety Gamma\\D\Gamma \backslash D.
2.5. Polarized Z\mathbb{Z}-variations of Hodge structures
Hodge theory as recalled in Section 2.1 can be considered as the particular case over a point of Hodge theory over an arbitrary base. Again, the motivation comes from geometry. Let f:Y rarr Bf: Y \rightarrow B be a proper surjective complex analytic submersion from a connected Kähler manifold YY to a complex manifold BB. It defines a locally constant sheaf V_(Z):=R^(∙)f_(**)Z\mathbb{V}_{\mathbb{Z}}:=R^{\bullet} f_{*} \mathbb{Z} of finitely generated abelian groups on BB, gathering the cohomologies H_(B)^(∙)(Y_(b),Z),b in BH_{\mathrm{B}}^{\bullet}\left(Y_{b}, \mathbb{Z}\right), b \in B. Upon choosing a base point b_(0)in Bb_{0} \in B, the datum of V_(Z)\mathbb{V}_{\mathbb{Z}} is equivalent to the datum of a monodromy representation rho:pi_(1)(B,b_(0))rarrGL(V_(Z,b_(0)))\rho: \pi_{1}\left(B, b_{0}\right) \rightarrow \mathbf{G L}\left(\mathbb{V}_{\mathbb{Z}, b_{0}}\right). On the other hand, the de Rham incarnation of the cohomology of the fibers of ff is the holomorphic flat vector bundle (V:=V_(Z)ox_(Z_(B))O_(B)≃:}\left(\mathcal{V}:=\mathbb{V}_{\mathbb{Z}} \otimes_{\mathbb{Z}_{B}} \mathcal{O}_{B} \simeq\right.{:R^(∙)f_(**)Omega_(Y//B)^(∙),grad)\left.R^{\bullet} f_{*} \Omega_{Y / B}^{\bullet}, \nabla\right), where O_(B)\mathcal{O}_{B} is the sheaf of holomorphic functions on B,Omega_(Y//B)^(∙)B, \Omega_{Y / B}^{\bullet} is the relative holomorphic de Rham complex and grad\nabla is the Gauss-Manin connection. The Hodge filtration on each H_(B)^(∙)(Y_(b),C)H_{\mathrm{B}}^{\bullet}\left(Y_{b}, \mathbb{C}\right) is induced by the holomorphic subbundles F^(p):=R^(∙)f_(**)Omega_(Y//B)^(∙ >= p)F^{p}:=R^{\bullet} f_{*} \Omega_{Y / B}^{\bullet \geq p} of V\mathcal{V}. The Hodge filtration is usually not preserved by the connection, but Griffiths [42] crucially observed that it satisfies the transversality constraint gradF^(p)subOmega_(B)^(1)ox_(O_(B))F^(p-1)\nabla F^{p} \subset \Omega_{B}^{1} \otimes_{\mathcal{O}_{B}} F^{p-1}. More generally, a variation of Z\mathbb{Z}-Hodge structures ( ZVHS\mathbb{Z} V H S ) on a connected complex manifold (B,O_(B))\left(B, \mathcal{O}_{B}\right) is a pair V:=(V_(Z),F^(∙))\mathbb{V}:=\left(\mathbb{V}_{\mathbb{Z}}, F^{\bullet}\right), consisting of a locally constant sheaf of finitely gener
ated abelian groups V_(Z)\mathbb{V}_{\mathbb{Z}} on BB and a (decreasing) filtration F^(∙)F^{\bullet} of the holomorphic vector bundle V:=V_(Z)ox_(Z_(B))O_(B)\mathcal{V}:=\mathbb{V}_{\mathbb{Z}} \otimes_{\mathbb{Z}_{B}} \mathcal{O}_{B} by holomorphic subbundles, called the Hodge filtration, satisfying the following conditions: for each b in Bb \in B, the pair (V_(b),F_(b)^(∙))\left(\mathbb{V}_{b}, F_{b}^{\bullet}\right) is a Z\mathbb{Z}-Hodge structure; and the flat connection grad\nabla on V\mathcal{V} defined by V_(C)\mathbb{V}_{\mathbb{C}} satisfies Griffiths' transversality,
A morphism VrarrV^(')\mathbb{V} \rightarrow \mathbb{V}^{\prime} of ZVHSs\mathbb{Z V H S s} on BB is a morphism f:V_(Z)rarrV_(Z)^(')f: \mathbb{V}_{\mathbb{Z}} \rightarrow \mathbb{V}_{\mathbb{Z}}^{\prime} of local systems such that the associated morphism of vector bundles f:VrarrV^(')f: \mathcal{V} \rightarrow \mathcal{V}^{\prime} is compatible with the Hodge filtrations. If V\mathbb{V} has weight kk, a polarization of V\mathbb{V} is a morphism q:VoxVrarrZ_(B)(-k)\mathrm{q}: \mathbb{V} \otimes \mathbb{V} \rightarrow \mathbb{Z}_{B}(-k) inducing a polarization on each Z\mathbb{Z}-Hodge structure V_(b),b in B\mathbb{V}_{b}, b \in B. In the geometric situation, such a polarization exists if there exists an element eta inH^(2)(Y,Z)\eta \in H^{2}(Y, \mathbb{Z}) whose restriction to each fiber Y_(b)Y_{b} defines a Kähler class, for instance if ff is the analytification of a smooth projective morphism of smooth connected algebraic varieties over C\mathbb{C}.
2.6. Generic Hodge datum and period map
Let SS be a smooth connected quasiprojective variety over C\mathbb{C} and let V\mathbb{V} be a polarized ZVHS\mathbb{Z V H S} on S^("an ")S^{\text {an }}. Fix a base point o inS^("an ")o \in S^{\text {an }}, let p: widetilde(S^("an "))rarrS^("an ")p: \widetilde{S^{\text {an }}} \rightarrow S^{\text {an }} be the corresponding universal cover and write V_(Z):=V_(Z,o),q_(Z):=q_(Z,o)V_{\mathbb{Z}}:=\mathbb{V}_{\mathbb{Z}, o}, q_{\mathbb{Z}}:=\mathrm{q}_{\mathbb{Z}, o}. The pulled-back polarized ZVHSp^(**)V\mathbb{Z} V H S p^{*} \mathbb{V} is canonically trivialized as (( widetilde(S^(an)))xxV_(Z),(( widetilde(S^("an ")))xxV_(C),F^(∙)),grad=d,q_(Z))\left(\widetilde{S^{\mathrm{an}}} \times V_{\mathbb{Z}},\left(\widetilde{S^{\text {an }}} \times V_{\mathbb{C}}, F^{\bullet}\right), \nabla=d, q_{\mathbb{Z}}\right). In [31, 7.5], Deligne proved that there exists a reductive Q\mathbb{Q}-algebraic subgroup iota:G↪GL(V_(Q))\iota: \mathbf{G} \hookrightarrow \mathbf{G L}\left(V_{\mathbb{Q}}\right), called the generic Mumford-
contained in G\mathbf{G}, and is equal to G\mathbf{G} outside of a meagre set of widetilde(S^("an "))\widetilde{S^{\text {an }}} (such points tilde(s)\tilde{s} are said Hodge generic for V\mathbb{V} ). A closed irreducible subvariety Y sub SY \subset S is said Hodge generic for V\mathbb{V} if it contains a Hodge generic point. The setup of Section 2.3 is thus in force. Without loss of generality, we can assume that the point tilde(o)\tilde{o} is Hodge generic. Let (G,D)(\mathbf{G}, D) be the Hodge datum (called the generic Hodge datum of S^("an ")S^{\text {an }} for V\mathbb{V} ) associated with the polarized Hodge structure (V_(Z),F_( tilde(o))^(∙))\left(V_{\mathbb{Z}}, F_{\tilde{o}}^{\bullet}\right). The ZVHSp^(**)V\mathbb{Z} V H S p^{*} \mathbb{V} is completely described by a holomorphic map widetilde(Phi): widetilde(S^(an))rarr D\widetilde{\Phi}: \widetilde{S^{\mathrm{an}}} \rightarrow D, which is naturally equivariant under the monodromy representation rho:pi_(1)(S^("an "),o)rarr Gamma:=G nnGL(V_(Z))\rho: \pi_{1}\left(S^{\text {an }}, o\right) \rightarrow \Gamma:=G \cap \mathbf{G L}\left(V_{\mathbb{Z}}\right), hence descends to a holomorphic map Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash D, called the period map of SS for V\mathbb{V}. We thus obtain the following commutative diagram in the category of complex analytic spaces:
Notice that the pair (V_(Q),(V,F^(∙)))\left(\mathbb{V}_{\mathbb{Q}},\left(\mathcal{V}, F^{\bullet}\right)\right) is the pullback under Phi\Phi of the pair (V_(iota),(V_(iota),F^(∙)))\left(\mathbb{V}_{\iota},\left(\mathcal{V}_{\iota}, F^{\bullet}\right)\right) on the Hodge variety Gamma\\D\Gamma \backslash D defined by the inclusion iota:G↪GL(V_(Q))\iota: \mathbf{G} \hookrightarrow \mathbf{G L}\left(V_{\mathbb{Q}}\right). Griffiths' transversality condition is equivalent to the statement that Phi\Phi is horizontal, d Phi(TS^(an))subT_(h)(Gamma\\D)d \Phi\left(T S^{\mathrm{an}}\right) \subset T_{h}(\Gamma \backslash D). By extension we call period map any holomorphic, horizontal, locally liftable map from S^("an ")S^{\text {an }} to a Hodge variety Gamma\\D\Gamma \backslash D.
The Hodge locus HL(S,V^(ox))\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right) of SS for V\mathbb{V} is the subset of points s inS^("an ")s \in S^{\text {an }} for which the Mumford-Tate group G_(s)\mathbf{G}_{s} is a strict subgroup of G\mathbf{G}, or equivalently for which the Hodge structure V_(s)\mathbb{V}_{s} admits more Hodge tensors than the very general fiber V_(s^('))\mathbb{V}_{s^{\prime}}. Thus
where the union is over all strict Hodge subdata and Gamma^(')\\D^(')\Gamma^{\prime} \backslash D^{\prime} is a slight abuse of notation for denoting the projection of D^(')sub DD^{\prime} \subset D to Gamma\\D\Gamma \backslash D.
Let Y sub SY \subset S be a closed irreducible algebraic subvariety i:Y↪Si: Y \hookrightarrow S. Let (G_(Y),D_(Y))\left(\mathbf{G}_{Y}, D_{Y}\right) be the generic Hodge datum of the ZVHSV\mathbb{Z} V H S \mathbb{V} restricted to the smooth locus of YY. The algebraic monodromy group H_(Y)\mathbf{H}_{Y} of YY for V\mathbb{V} is the identity component of the Zariski-closure in GL(V_(Q))\mathbf{G L}\left(V_{\mathbb{Q}}\right) of the monodromy of the restriction to YY of the local system V_(Z)\mathbb{V}_{\mathbb{Z}}. It follows from Deligne's (in the geometric case) and Schmid's (in general) "Theorem of the fixed part" and "Semisimplicity Theorem" that H_(Y)\mathbf{H}_{Y} is a normal subgroup of the derived group G_(Y)^(der)\mathbf{G}_{Y}^{\mathrm{der}}, see [2, THEOREM 1].
3. HODGE THEORY AND TAME GEOMETRY
3.1. Variational Hodge theory between algebraicity and transcendence
Let SS be a smooth connected quasi-projective variety over C\mathbb{C} and let V=(V_(Z),F^(∙))\mathbb{V}=\left(\mathbb{V}_{\mathbb{Z}}, F^{\bullet}\right) be a polarized ZVHS\mathbb{Z V H S} on S^("an ")S^{\text {an }}. Let (G,D)(\mathbf{G}, D) be the generic Hodge datum of SS for V\mathbb{V} and let Phi:S^(an)rarr Gamma\\D\Phi: S^{\mathrm{an}} \rightarrow \Gamma \backslash D be the period map defined by V\mathbb{V}.
The fact that Hodge theory is a transcendental theory is reflected in the following facts:
First, the triplets (V_(lambda),(V_(lambda),F^(∙)),grad)\left(\mathbb{V}_{\lambda},\left(\mathcal{V}_{\lambda}, F^{\bullet}\right), \nabla\right) on Gamma\\D\Gamma \backslash D (for lambda:GrarrGL(V_(Q))\lambda: \mathbf{G} \rightarrow \mathbf{G L}\left(V_{\mathbb{Q}}\right) an algebraic representation) do not in general satisfy Griffiths' transversality, hence do not define a ZVHS\mathbb{Z V H S} on Gamma\\D\Gamma \backslash D. They do if and only if V\mathbb{V} is of Shimura type, i.e., (G,D)(\mathbf{G}, D) is a (connected) Shimura datum (meaning that the weight zero Hodge structures on the fibers of V_("Ad ")\mathbb{V}_{\text {Ad }} are of type {:{(-1,1),(0,0),(1,-1)})\left.\{(-1,1),(0,0),(1,-1)\}\right); or equivalently, if the horizontal tangent bundle T_(h)DT_{h} D coincides with TDT D. In other words, Hodge varieties are in general not classifying spaces for polarized ZVHS\mathbb{Z V H S}.
Second, and more importantly, the complex analytic Hodge variety Gamma\\D\Gamma \backslash D is in general not algebraizable (i.e., it is not the analytification of a complex quasiprojective variety). More precisely, let us write D=G//MD=G / M as in Section 2.3. A classical property of elliptic orbits like DD is that there exists a unique maximal compact subgroup KK of GG containing MM [46]. Supposing for simplicity that GG is a real simple Lie group GG, then Gamma\\D\Gamma \backslash D is algebraizable only if G//KG / K is a hermitian symmetric domain and the projection D rarr G//KD \rightarrow G / K is holomorphic or antiholomorphic, see [45][45].
On the other hand, this transcendence is severely constrained, as shown by the following algebraicity results:
If (G,D)(\mathbf{G}, D) is of Shimura type then Gamma\\D=Sh^(an)\Gamma \backslash D=\mathrm{Sh}^{\mathrm{an}} is the analytification of an algebraic variety, called a Shimura variety Sh [8,30,32]. In that case Borel [17, THEOREM 3.10] proved that the complex analytic period map Phi:S^(an)rarrSh^("an ")\Phi: S^{\mathrm{an}} \rightarrow \mathrm{Sh}^{\text {an }} is the analytification of an algebraic map.
Let S sub bar(S)S \subset \bar{S} be a log-smooth compactification of SS by a simple normal crossing divisor ZZ. Following Deligne [28], the flat holomorphic connection grad\nabla on V\mathcal{V} defines a canonical extension bar(V)\overline{\mathcal{V}} of V\mathcal{V} to bar(S)\bar{S}. Using GAGA for bar(S)\bar{S}, this defines an algebraic structure on (V,grad)(\mathcal{V}, \nabla), for which the connection grad\nabla is regular. Around any point of ZZ, the complex manifold S^("an ")S^{\text {an }} is locally isomorphic to a product (Delta^(**))^(k)xxDelta^(l)\left(\Delta^{*}\right)^{k} \times \Delta^{l} of punctured polydisks. Borel showed that the monodromy representation rho:pi_(1)(S^("an "),s_(o))rarr Gamma subG(Q)\rho: \pi_{1}\left(S^{\text {an }}, s_{o}\right) \rightarrow \Gamma \subset \mathbf{G}(\mathbb{Q}) of V\mathbb{V} is "tame at infinity," that is, its restriction to Z^(k)=pi_(1)((Delta^(**))^(k)xxDelta^(l))\mathbb{Z}^{k}=\pi_{1}\left(\left(\Delta^{*}\right)^{k} \times \Delta^{l}\right) is quasiunipotent, see [82, (4.5)]. Using this result, Schmid showed that the Hodge filtration F^(∙)F^{\bullet} extends holomorphically to the Deligne extension bar(V)\overline{\mathcal{V}}. This is the celebrated Nilpotent Orbit theorem [82, (4.12)]. It follows, as noticed by Griffiths [82, (4.13)], that the Hodge filtration on V\mathcal{V} comes from an algebraic filtration on the underlying algebraic bundle, whether V\mathbb{V} is of geometric origin or not.
More recently, an even stronger evidence came from the study of Hodge loci. Cattani, Deligne, and Kaplan proved the following celebrated result (generalized to the mixed case in [18-21]):
Theorem 3.1 ([22]). Let SS be a smooth connected quasiprojective variety over C\mathbb{C} and V\mathbb{V} be a polarized ZVHS\mathbb{Z} V H S over SS. Then HL(S,V^(ox))\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right) is a countable union of closed irreducible algebraic subvarieties of SS.
In view of this tension between algebraicity and transcendence, it is natural to ask if there is a framework, less strict than complex algebraic geometry but more constraining than complex analytic geometry, where to analyze period maps and explain its remarkable properties.
3.2. O-minimal geometry
Such a framework was in fact envisioned by Grothendieck in [47, 85] under the name "tame topology," as a way out of the pathologies of general topological spaces. Examples of pathologies are Cantor sets, space-filling curves but also much simpler objects like the graph Gamma:={(x,sin ((1)/(x))),0 < x <= 1}subR^(2):\Gamma:=\left\{\left(x, \sin \frac{1}{x}\right), 0<x \leq 1\right\} \subset \mathbb{R}^{2}: its closure bar(Gamma):=Gamma⨿I\bar{\Gamma}:=\Gamma \amalg \mathrm{I}, where I:={0}xx[-1,1]subR^(2)\mathrm{I}:=\{0\} \times[-1,1] \subset \mathbb{R}^{2} is connected but not arc-connected; dim( bar(Gamma)\\Gamma)=dim Gamma\operatorname{dim}(\bar{\Gamma} \backslash \Gamma)=\operatorname{dim} \Gamma, which prevents any reasonable stratification theory; and Gamma nnR\Gamma \cap \mathbb{R} is not "of finite type." Tame geometry has been developed by model theorists as o-minimal geometry, which studies structures where every definable set has a finite geometric complexity. Its prototype is real semialgebraic geometry, but it is much richer. We refer to [34] for a nice survey.
Definition 3.2. A structure SS expanding the real field is a collection S=(S_(n))_(n inN)S=\left(S_{n}\right)_{n \in \mathbb{N}}, where S_(n)S_{n} is a set of subsets of R^(n)\mathbb{R}^{n} such that for every n inNn \in \mathbb{N} :
(1) all algebraic subsets of R^(n)\mathbb{R}^{n} are in S_(n)S_{n}.
(2) S_(n)S_{n} is a boolean subalgebra of the power set of R^(n)\mathbb{R}^{n} (i.e., S_(n)S_{n} is stable by finite union, intersection, and complement).
(3) If A inS_(n)A \in S_{n} and B inS_(m)B \in S_{m} then A xx B inS_(n+m)A \times B \in S_{n+m}.
(4) Let p:R^(n+1)rarrR^(n)p: \mathbb{R}^{n+1} \rightarrow \mathbb{R}^{n} be a linear projection. If A inS_(n+1)A \in S_{n+1} then p(A)inS_(n)p(A) \in S_{n}.
The elements of S_(n)S_{n} are called the S\mathcal{S}-definable sets of R^(n)\mathbb{R}^{n}. A map f:A rarr Bf: A \rightarrow B between S\mathcal{S} definable sets is said to be S\mathcal{S}-definable if its graph is S\mathcal{S}-definable.
A dual point of view starts from the functions, namely considers sets definable in a first-order structure S=(:R,+,xx, < ,(f_(i))_(i in I):)S=\left\langle\mathbb{R},+, \times,<,\left(f_{i}\right)_{i \in I}\right\rangle where II is a set and the f_(i):R^(n_(i))rarrR,i in If_{i}: \mathbb{R}^{n_{i}} \rightarrow \mathbb{R}, i \in I, are functions. A subset Z subR^(n)Z \subset \mathbb{R}^{n} is SS-definable if it can be defined by a formula
Z:={(x_(1),dots,x_(n))inR^(n)∣phi(x_(1),dots,x_(n))" is true "}Z:=\left\{\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n} \mid \phi\left(x_{1}, \ldots, x_{n}\right) \text { is true }\right\}
where phi\phi is a first-order formula that can be written using only the quantifiers AA\forall and EE\exists applied to real variables; logical connectors; algebraic expressions written with the f_(i)f_{i}; the order symbol <<; and fixed parameters lambda_(i)inR\lambda_{i} \in \mathbb{R}. When the set II is empty the SS-definable subsets are the semialgebraic sets. Semialgebraic subsets are thus always SS-definable.
One easily checks that the composite of delta\delta-definable functions is delta\delta-definable, as are the images and the preimages of delta\delta-definable sets under delta\delta-definable maps. Using that the euclidean distance is a real-algebraic function, one shows easily that the closure and interior of an S\mathcal{S}-definable set are again S\mathcal{S}-definable.
The following o-minimal axiom for a structure SS guarantees the possibility of doing geometry using S\mathcal{S}-definable sets as basic blocks.
Definition 3.3. A structure SS is said to be o-minimal if S_(1)S_{1} consists precisely of the finite unions of points and intervals (i.e., the semialgebraic subsets of R\mathbb{R} ).
Example 3.4. The structure R_(sin):=(:R,+,xx, < ,sin:)\mathbb{R}_{\sin }:=\langle\mathbb{R},+, \times,<, \sin \rangle is not o-minimal. Indeed, the infinite union of points piZ={x inR∣sin x=0}\pi \mathbb{Z}=\{x \in \mathbb{R} \mid \sin x=0\} is a definable subset of R\mathbb{R} in this structure.
Any o-minimal structure SS has the following main tameness property: given finitely many SS-definable sets U_(1),dots,U_(k)subR^(n)U_{1}, \ldots, U_{k} \subset \mathbb{R}^{n}, there exists a definable cylindrical cellular decomposition of R^(n)\mathbb{R}^{n} such that each U_(i)U_{i} is a finite union of cells. Such a decomposition is defined inductively on nn. For n=1n=1, this is a finite partition of R\mathbb{R} into cells which are points or open intervals. For n > 1n>1, it is obtained from a definable cylindrical cellular decomposition of R^(n-1)\mathbb{R}^{n-1} by fixing, for any cell C subR^(n-1)C \subset \mathbb{R}^{n-1}, finitely many definable functions f_(C,i):C rarrRf_{C, i}: C \rightarrow \mathbb{R}, 1 <= i <= k_(C)1 \leq i \leq k_{C}, with f_(C,0):=-oo < f_(C,1) < cdots < f_(C,k_(C)) < f_(C,k_(C)+1):=+oof_{C, 0}:=-\infty<f_{C, 1}<\cdots<f_{C, k_{C}}<f_{C, k_{C}+1}:=+\infty, and defining the cells of R^(n)\mathbb{R}^{n} as the graphs {(x,f_(C,i)(x)),x in C},1 <= i <= k_(C)\left\{\left(x, f_{C, i}(x)\right), x \in C\right\}, 1 \leq i \leq k_{C}, and the bands {(x,f_(C,i)(x) < :}\left\{\left(x, f_{C, i}(x)<\right.\right.{:y < f_(C,i+1)(x)),x in C,y inR},0 <= i <= k_(C)\left.\left.y<f_{C, i+1}(x)\right), x \in C, y \in \mathbb{R}\right\}, 0 \leq i \leq k_{C}, for all cells CC of R^(n-1)\mathbb{R}^{n-1}.
The simplest o-minimal structure is the structure R_("alg ")\mathbb{R}_{\text {alg }} consisting of semialgebraic sets. It is too close to algebraic geometry to be used for studying transcendence phenomena. Luckily much richer o-minimal geometries do exist. A fundamental result of Wilkie, building on the result of Khovanskii [54] that any exponential set {(x_(1),dots,x_(n))in:}\left\{\left(x_{1}, \ldots, x_{n}\right) \in\right.{:R^(n)∣P(x_(1),dots,x_(n),exp(x_(1)),dots,exp(x_(n)))=0}(:}\left.\mathbb{R}^{n} \mid P\left(x_{1}, \ldots, x_{n}, \exp \left(x_{1}\right), \ldots, \exp \left(x_{n}\right)\right)=0\right\}\left(\right. where {:P inR[X_(1),dots,X_(n),Y_(1),dots,Y_(n)])\left.P \in \mathbb{R}\left[X_{1}, \ldots, X_{n}, Y_{1}, \ldots, Y_{n}\right]\right) has finitely many connected components, states:
Theorem 3.5 ([97]). The structure R_(exp):=(:R,+,xx, < ,exp:RrarrR:)\mathbb{R}_{\exp }:=\langle\mathbb{R},+, \times,<, \exp : \mathbb{R} \rightarrow \mathbb{R}\rangle is o-minimal.
In another direction, let us define
R_(an):=(:R,+,xx, < ,{f}" for "f" restricted real analytic function ":)\mathbb{R}_{\mathrm{an}}:=\langle\mathbb{R},+, \times,<,\{f\} \text { for } f \text { restricted real analytic function }\rangle
where a function f:R^(n)rarrRf: \mathbb{R}^{n} \rightarrow \mathbb{R} is a restricted real analytic function if it is zero outside [0,1]^(n)[0,1]^{n} and if there exists a real analytic function gg on a neighborhood of [0,1]^(n)[0,1]^{n} such that ff and gg are equal on [0,1]^(n)[0,1]^{n}. Gabrielov's result [37] that the difference of two subanalytic sets is subanalytic implies rather easily that the structure R_(an)\mathbb{R}_{\mathrm{an}} is o-minimal. The structure generated by two o-minimal structures is not o-minimal in general, but Van den Dries and Miller [35] proved that the structure R_("an,exp ")\mathbb{R}_{\text {an,exp }} generated by R_("an ")\mathbb{R}_{\text {an }} and R_("exp ")\mathbb{R}_{\text {exp }} is o-minimal. This is the ominimal structure which will be mainly used in the rest of this text.
Let us now globalize the notion of definable set using charts:
Definition 3.6. A definable topological space XX is the data of a Hausdorff topological space X\mathcal{X}, a finite open covering (U_(i))_(1 <= i <= k)\left(U_{i}\right)_{1 \leq i \leq k} of X\mathcal{X}, and homeomorphisms psi_(i):U_(i)rarrV_(i)subR^(n)\psi_{i}: U_{i} \rightarrow V_{i} \subset \mathbb{R}^{n} such that all V_(i),V_(ij):=psi_(i)(U_(i)nnU_(j))V_{i}, V_{i j}:=\psi_{i}\left(U_{i} \cap U_{j}\right) and psi_(i)@psi_(j)^(-1):V_(ij)rarrV_(ji)\psi_{i} \circ \psi_{j}^{-1}: V_{i j} \rightarrow V_{j i} are definable. As usual the pairs (U_(i),psi_(i))\left(U_{i}, \psi_{i}\right) are called charts. A morphism of definable topological spaces is a continuous map which is definable when read in the charts. The definable site X__(X)\underline{X}_{\mathcal{X}} of a definable topological space X\mathcal{X} has for objects definable open subsets U sub XU \subset X and admissible coverings are the finite ones.
Example 3.7. Let XX be an algebraic variety over R\mathbb{R}. Then X(R)X(\mathbb{R}) equipped with the euclidean topology carries a natural R_("alg ")\mathbb{R}_{\text {alg }}-definable structure (up to isomorphism): one covers XX by finitely many (Zariski) open affine subvarieties X_(i)X_{i} and take U_(i):=X_(i)(R)U_{i}:=X_{i}(\mathbb{R}) which is naturally a semialgebraic set. One easily check that any two finite open affine covers define isomorphic R_("alg ")\mathbb{R}_{\text {alg }}-structures on X(R)X(\mathbb{R}). If XX is an algebraic variety over C\mathbb{C} then X(C)=(Res_(C)//RX)(R)X(\mathbb{C})=\left(\operatorname{Res}_{\mathbb{C}} / \mathbb{R} X\right)(\mathbb{R}) carries thus a natural R_("alg ")\mathbb{R}_{\text {alg }}-structure. We call this the R_("alg ")\mathbb{R}_{\text {alg }}-definabilization of XX and denote it by X^(R_("alg "))X^{\mathbb{R}_{\text {alg }}}.
In the rest of this section, we fix an o-minimal structure SS and write "definable" for S\mathcal{S}-definable. Given a complex algebraic variety XX we write X^("def ")X^{\text {def }} for the S\mathcal{S}-definabilization X^(S)X^{\mathcal{S}}.
3.3. O-minimal geometry and algebraization
Why should an algebraic geometer care about o-minimal geometry? Because ominimal geometry provides strong algebraization results.
3.3.1. Diophantine criterion
The first algebraization result is the celebrated Pila-Wilkie theorem:
Theorem 3.8 ([77]). Let Z subR^(n)Z \subset \mathbb{R}^{n} be a definable set. We define Z^("alg ")Z^{\text {alg }} as the union of all connected positive-dimensional semialgebraic subsets of ZZ. Then, denoting by H:Q^(n)rarrRH: \mathbb{Q}^{n} \rightarrow \mathbb{R} the standard height function:
In words, if a definable set contains at least polynomially many rational points (with respect to their height), then it contains a positive dimensional semialgebraic set! For instance, if f:RrarrRf: \mathbb{R} \rightarrow \mathbb{R} is a real analytic function such that its graph Gamma_(f)nn[0,1]xx[0,1]\Gamma_{f} \cap[0,1] \times[0,1] contains at least polynomially many rational points (with respect to their height), then the function ff is real algebraic [15]. This algebraization result is a crucial ingredient in the proof of functional transcendence results for period maps, see Section 4.
3.3.2. Definable Chow and definable GAGA
In another direction, algebraicity follows from the meeting of o-minimal geometry with complex geometry. The motto is that o-minimal geometry is incompatible with the many pathologies of complex analysis. As a simple illustration, let f:Delta^(**)rarrCf: \Delta^{*} \rightarrow \mathbb{C} be a holomorphic function, and assume that ff is definable (where we identify C\mathbb{C} with R^(2)\mathbb{R}^{2} and Delta^(**)subR^(2)\Delta^{*} \subset \mathbb{R}^{2} is semialgebraic). Then ff does not have any essential singularity at 0 (i.e., ff is meromorphic). Otherwise, by the Big Picard theorem, the boundary bar(Gamma_(f))\\Gamma_(f)\overline{\Gamma_{f}} \backslash \Gamma_{f} of its graph would contain {0}xxC\{0\} \times \mathbb{C}, hence would have the same real dimension (two) as Gamma_(f)\Gamma_{f}, contradicting the fact that Gamma_(f)\Gamma_{f} is definable.
Let us first define a good notion of a definable topological space "endowed with a complex analytic structure." We identify C^(n)\mathbb{C}^{n} with R^(2n)\mathbb{R}^{2 n} by taking real and imaginary parts. Given U subC^(n)U \subset \mathbb{C}^{n} a definable open subset, let O_(C^(n))(U)\mathcal{O}_{\mathbb{C}^{n}}(U) denote the C\mathbb{C}-algebra of holomorphic definable functions U rarrCU \rightarrow \mathbb{C}. The assignment Uâ‡O_(C^(n))(U)U \rightsquigarrow \mathcal{O}_{\mathbb{C}^{n}}(U) defines a sheaf O_(C^(n))\mathcal{O}_{\mathbb{C}^{n}} on C^(n)\mathbb{C}^{n} whose stalks are local rings. Given a finitely generated ideal I subO_(C^(n))(U)I \subset \mathcal{O}_{\mathbb{C}^{n}}(U), its zero locus V(I)sub UV(I) \subset U is definable and the restriction O_(V(I)):=(O_(U)//IO_(U))_(V(I)_)\mathcal{O}_{V(I)}:=\left(\mathcal{O}_{U} / I \mathcal{O}_{U}\right)_{\underline{V(I)}} define a sheaf of local rings on V(I)_\underline{V(I)}.
Definition 3.9. A definable complex analytic space is a pair (X,O_(X))\left(\mathcal{X}, \mathcal{O}_{X}\right) consisting of a definable topological space X\mathcal{X} and a sheaf O_(X)\mathcal{O}_{X} on X_\underline{X} such that there exists a finite covering of X\mathcal{X} by definable open subsets X_(i)\mathcal{X}_{i} on which (X,O_(X))_(∣X_(i))\left(\mathcal{X}, \mathcal{O}_{X}\right){ }_{\mid X_{i}} is isomorphic to some (V(I),O_(V(I)))\left(V(I), \mathcal{O}_{V(I)}\right).
Bakker et al. [10, THEOREM 2.16] show that this is a reasonable definition: the sheaf O_(X)\mathcal{O}_{X}, in analogy with the classical Oka's theorem, is a coherent sheaf of rings. Moreover, one has a natural definabilization functor (X,O_(X))â‡(X^("def "),O_(X^("def ")))\left(X, \mathcal{O}_{X}\right) \rightsquigarrow\left(X^{\text {def }}, \mathcal{O}_{X^{\text {def }}}\right) from the category of separated schemes (or algebraic spaces) of finite type over C\mathbb{C} to the category of definable complex analytic spaces, which induces a morphism g:(X^("def ")_,O_(X^("def ")))rarr(X_,O_(X))g:\left(\underline{X^{\text {def }}}, \mathcal{O}_{X^{\text {def }}}\right) \rightarrow\left(\underline{X}, \mathcal{O}_{X}\right) of locally ringed sites.
Let us now describe the promised algebraization results. The classical Chow's theorem states that a closed complex analytic subset ZZ of X^("an ")X^{\text {an }} for XX smooth projective over C\mathbb{C} is in fact algebraic. This fails dramatically if XX is only quasiprojective, as shown by the graph of the complex exponential in (A^(2))^("an ")\left(\mathbb{A}^{2}\right)^{\text {an }}. However, Peterzil and Starchenko, generalizing [36] in the R_("alg ")\mathbb{R}_{\text {alg }} case, have shown the following:
Theorem 3.10 ([69,70]). Let XX be a complex quasiprojective variety and let Z subX^("an ")Z \subset X^{\text {an }} be a closed analytic subvariety. If ZZ is definable in X^("def ")X^{\text {def }} then ZZ is complex algebraic in XX.
Chow's theorem, which deals only with spaces, was extended to sheaves by Serre [83]: when XX is proper, the analytification functor (*)^("an "):Coh(X)rarr Coh(X^("an "))(\cdot)^{\text {an }}: \operatorname{Coh}(X) \rightarrow \operatorname{Coh}\left(X^{\text {an }}\right) defines an equivalence of categories between the categories of coherent sheaves Coh(X)\operatorname{Coh}(X) and Coh(X^(an))\operatorname{Coh}\left(X^{\mathrm{an}}\right). In the definable world, let XX be a separated scheme (or algebraic space) of finite type over C\mathbb{C}. Associating with a coherent sheaf FF on XX the coherent sheaf F^("def "):=Fox_(g^(-1))O_(X)O_(X^("def "))F^{\text {def }}:=F \otimes_{g^{-1}} \mathcal{O}_{X} \mathcal{O}_{X^{\text {def }}} on the SS-definabilization X^("def ")X^{\text {def }} of XX, one obtains a definabilization functor (.) )^("def "):Coh(X)rarr)^{\text {def }}: \operatorname{Coh}(X) \rightarrowCoh(X^("def "))\operatorname{Coh}\left(X^{\text {def }}\right). Similarly there is an analytification functor Xâ‡X^("an ")X \rightsquigarrow X^{\text {an }} from complex definable analytic spaces to complex analytic spaces, that induces a functor (*)^(an):Coh(X)rarr(\cdot)^{\mathrm{an}}: \operatorname{Coh}(\mathcal{X}) \rightarrowCoh(X^(an))\operatorname{Coh}\left(\mathcal{X}^{\mathrm{an}}\right).
Theorem 3.11 ([10]). For every separated algebraic space of finite type XX, the definabilization functor (*)^(def):Coh(X)rarr Coh(X^(def))(\cdot)^{\mathrm{def}}: \operatorname{Coh}(X) \rightarrow \operatorname{Coh}\left(X^{\mathrm{def}}\right) is exact and fully faithful (but it is not necessarily essentially surjective). Its essential image is stable under subobjects and subquotients.
Using Theorem 3.11 and Artin's algebraization theorem for formal modification [4], one obtains the following useful algebraization result for definable images of algebraic spaces, which will be used in Section 3.6.2:
Theorem 3.12 ([10]). Let XX be a separated algebraic space of finite type and let E\mathcal{E} be a definable analytic space. Any proper definable analytic map Phi:X^("def ")rarrE\Phi: X^{\text {def }} \rightarrow \mathcal{E} factors uniquely as ∙@f^(def)\bullet \circ f^{\mathrm{def}}, where f:X rarr Yf: X \rightarrow Y is a proper morphism of separated algebraic spaces (of finite type) such that O_(Y)rarrf_(**)O_(X)\mathcal{O}_{Y} \rightarrow f_{*} \mathcal{O}_{X} is injective, and iota:Y^("def ")↪E\iota: Y^{\text {def }} \hookrightarrow \mathcal{E} is a closed immersion of definable analytic spaces.
3.4. Definability of Hodge varieties
Let us now describe the first result establishing that o-minimal geometry is potentially interesting for Hodge theory.
Theorem 3.13 ([11]). Any Hodge variety Gamma\\D\Gamma \backslash D can be naturally endowed with a functorial structure (Gamma\\D)^(R_("alg "))(\Gamma \backslash D)^{\mathbb{R}_{\text {alg }}} of R_("alg ")\mathbb{R}_{\text {alg }}-definable complex analytic space.
Here "functorial" means that that any morphism (G^('),D^('))rarr(G,D)\left(\mathbf{G}^{\prime}, D^{\prime}\right) \rightarrow(\mathbf{G}, D) of Hodge data induces a definable map (Gamma^(')\\D^('))^(R_("alg "))rarr(Gamma\\D)^(R_("alg "))\left(\Gamma^{\prime} \backslash D^{\prime}\right)^{\mathbb{R}_{\text {alg }}} \rightarrow(\Gamma \backslash D)^{\mathbb{R}_{\text {alg }}} of Hodge varieties. Let us sketch the construction of (Gamma\\D)^(R_("alg "))(\Gamma \backslash D)^{\mathbb{R}_{\text {alg }}}. Without loss of generality (replacing G\mathbf{G} by its adjoint group if necessary), we can assume that G\mathbf{G} is semisimple, G=G(R)^(+)G=\mathbf{G}(\mathbb{R})^{+}. For simplicity, let us assume that the arithmetic lattice Gamma\Gamma is torsion free. We choose a base point in D=G//MD=G / M. Notice that
Once Theorem 3.13 is in place, the following result shows that o-minimal geometry is a natural framework for Hodge theory:
Theorem 3.14 ([11]). Let SS be a smooth connected complex quasiprojective variety. Any period map Phi:S^(an)rarr Gamma\\D\Phi: S^{\mathrm{an}} \rightarrow \Gamma \backslash D is the analytification of a morphism Phi:S^(R_(an,exp))rarr(Gamma\\D)^(R_(an,exp))\Phi: S^{\mathbb{R}_{\mathrm{an}, \mathrm{exp}}} \rightarrow(\Gamma \backslash D)^{\mathbb{R}_{\mathrm{an}, \mathrm{exp}}} of R_(an,exp)\mathbb{R}_{\mathrm{an}, \exp }-definable complex analytic spaces, where the R_(an,exp)\mathbb{R}_{\mathrm{an}, \exp }-structures on S(C)S(\mathbb{C}) and Gamma\\D\Gamma \backslash D extend their natural R_("alg ")\mathbb{R}_{\text {alg }}-structures defined in Example 3.7 and Theorem 3.13, respectively.
In down-to-earth terms, this means that we can cover SS by finitely many open affine charts S_(i)S_{i} such that Phi\Phi restricted to (Res_(C//R)S_(i))(R)=S_(i)(C)\left(\operatorname{Res}_{\mathbb{C} / \mathbb{R}} S_{i}\right)(\mathbb{R})=S_{i}(\mathbb{C}) and read in a chart of Gamma\\D\Gamma \backslash D defined by a Siegel set of DD, can be written using only real polynomials, the real exponential function, and restricted real analytic functions! This statement is already nontrivial when S=S= Sh is a Shimura variety and Phi^(an):S^(an)rarr Gamma\\D\Phi^{\mathrm{an}}: S^{\mathrm{an}} \rightarrow \Gamma \backslash D is the identity map coming from the uniformization pi:D rarrS^("an ")\pi: D \rightarrow S^{\text {an }} of S^("an ")S^{\text {an }} by the hermitian symmetric domain D=G//KD=G / K. In that case the R_("alg ")\mathbb{R}_{\text {alg }}-definable varieties Sh^(R_("alg "))\mathrm{Sh}^{\mathbb{R}_{\text {alg }}} and (Gamma\\D)^(R_("alg "))(\Gamma \backslash D)^{\mathbb{R}_{\text {alg }}} are not isomorphic, but Theorem 3.14 claims that their R_(an,exp)\mathbb{R}_{\mathrm{an}, \exp }-extensions Sh^(R_(an,exp))\operatorname{Sh}^{\mathbb{R}_{\mathrm{an}, \mathrm{exp}}} and (Gamma\\D)^(R_(an," exp "))(\Gamma \backslash D)^{\mathbb{R}_{\mathrm{an}, \text { exp }}} are. This is equivalent to showing that the restriction pi_(∣ℑ_(D)):S_(D)rarrS^(R_(an," exp "))\pi_{\mid \Im_{D}}: \mathbb{S}_{D} \rightarrow S^{\mathbb{R}_{\mathrm{an}, \text { exp }}} to a Siegel set for DD can be written using only real polynomials, the real exponential function, and restricted real analytic functions. This
is a nice exercise on the jj-function when Sh is a modular curve, was done in [71] and [76] for Sh=A_(g)\mathrm{Sh}=\mathscr{A}_{g}, and [58] in general.
Let us sketch the proof of Theorem 3.14. We choose a log-smooth compactification of SS, hence providing us with a definable cover of S^(R_("an "))S^{\mathbb{R}_{\text {an }}} by punctured polydisks (Delta^(**))^(k)xxDelta^(l)\left(\Delta^{*}\right)^{k} \times \Delta^{l}. We are reduced to showing that the restriction of Phi\Phi to such a punctured polydisk is R_(an,exp^(-))\mathbb{R}_{\mathrm{an}, \exp ^{-}} definable. This is clear if k=0k=0, as in this case varphi:Delta^(k+l)rarr Gamma\\D\varphi: \Delta^{k+l} \rightarrow \Gamma \backslash D is even R_("an ")\mathbb{R}_{\text {an }}-definable. For k > 0k>0, let e:exp(2pii*):SrarrDelta^(**)\mathrm{e}: \exp (2 \pi \mathrm{i} \cdot): \mathfrak{S} \rightarrow \Delta^{*} be the universal covering map. Its restriction to a sufficiently large bounded vertical strip V:=[a,b]xx]0,+oo[subN={x+iy,y > 0}V:=[a, b] \times] 0,+\infty[\subset \mathfrak{N}=\{x+\mathrm{i} y, y>0\} is R_(an,exp)\mathbb{R}_{\mathrm{an}, \mathrm{exp}}-definable. Considering the following commutative diagram:
it is thus enough to show that pi@ widetilde(Phi):V^(k)xxDelta^(l)rarr Gamma\\D\pi \circ \widetilde{\Phi}: V^{k} \times \Delta^{l} \rightarrow \Gamma \backslash D is R_("an,exp ")\mathbb{R}_{\text {an,exp }}-definable.
i <= ki \leq k, and the variables t_(j),k+1 <= j <= k+lt_{j}, k+1 \leq j \leq k+l. On the other hand, exp(sum_(i=1)^(k)z_(i)N_(i))in\exp \left(\sum_{i=1}^{k} z_{i} N_{i}\right) \inG(C)\mathbf{G}(\mathbb{C}) is polynomial in the variables z_(i)z_{i}, as the monodromies N_(i)N_{i} are nilpotent and commute pairwise. As the action of G(C)\mathbf{G}(\mathbb{C}) on D^(ˇ)\check{D} is algebraic, it follows that widetilde(Phi):V^(k)xxDelta^(l)rarr D\widetilde{\Phi}: V^{k} \times \Delta^{l} \rightarrow D is R_(an,exp)\mathbb{R}_{\mathrm{an}, \exp }-definable. The proof of Theorem 3.14 is thus reduced to the following, proven by Schmid when k=1,l=0[82,5.29]k=1, l=0[82,5.29] :
Theorem 3.15 ([11]). The image widetilde(Phi)(V^(k)xxDelta^(l))\widetilde{\Phi}\left(V^{k} \times \Delta^{l}\right) lies in a finite union of Siegel sets of DD.
This can be interpreted as showing that, possibly after passing to a definable cover of V^(k)xxDelta^(l)V^{k} \times \Delta^{l}, the Hodge form of widetilde(Phi)\widetilde{\Phi} is Minkowski reduced with respect to a flat frame. This is done using the hard analytic theory of Hodge forms estimates for degenerations of variations of Hodge structure, as in [53, THEOREMS 3.4.1 AND 3.4.2] and [23, THEOREM 5.21].
Remark 3.16. Theorems 3.13 and 3.14 have been extended to the mixed case in [9].
3.6. Applications
3.6.1. About the Cattani-Deligne-Kaplan theorem
As a corollary of Theorems 3.14 and 3.10 one obtains the following, which, in view of (2.3), implies immediately Theorem 3.1 :
Theorem 3.17 ([11]). Let SS be a smooth quasiprojective complex variety. Let V\mathbb{V} be a polarized ZVHS\mathbb{Z} V H S on S^(an)S^{\mathrm{an}} with period map Phi:S^(an)rarr Gamma\\D\Phi: S^{\mathrm{an}} \rightarrow \Gamma \backslash D. For any special subvariety Gamma^(')\\D^(')sub\Gamma^{\prime} \backslash D^{\prime} \subsetGamma\\D\Gamma \backslash D, its preimage Phi^(-1)(Gamma^(')\\D^('))\Phi^{-1}\left(\Gamma^{\prime} \backslash D^{\prime}\right) is a finite union of irreducible algebraic subvarieties of SS.
Indeed, it follows from Theorem 3.13 that Gamma^(')\\D^(')\Gamma^{\prime} \backslash D^{\prime} is definable in (Gamma\\D)^(R_("alg "))(\Gamma \backslash D)^{\mathbb{R}_{\text {alg }}}. By Theorem 3.14, its preimage Phi^(-1)(Gamma^(')\\D^('))\Phi^{-1}\left(\Gamma^{\prime} \backslash D^{\prime}\right) is definable in S^(R_("annexp "))S^{\mathbb{R}_{\text {annexp }}}. As Phi\Phi is holomorphic and Gamma^(')\\D^(')sub Gamma\\D\Gamma^{\prime} \backslash D^{\prime} \subset \Gamma \backslash D is a closed complex analytic subvariety, Phi^(-1)(Gamma^(')\\D^('))\Phi^{-1}\left(\Gamma^{\prime} \backslash D^{\prime}\right) is also a closed complex analytic subvariety of S^("an ")S^{\text {an }}. By Theorem 3.10, it is thus algebraic in SS.
Remark 3.18. Theorem 3.17 has been extended to the mixed case in [9], thus recovering [18-21][18-21].
Let Y sub SY \subset S be a closed irreducible algebraic subvariety. Let (G_(Y),D_(Y))sub(G,D)\left(\mathbf{G}_{Y}, D_{Y}\right) \subset(\mathbf{G}, D) be the generic Hodge datum of V\mathbb{V} restricted to the smooth locus of of YY. There exist a smallest Hodge subvariety Gamma_(Y)\\D_(Y)\Gamma_{Y} \backslash D_{Y} of Gamma\\D\Gamma \backslash D containing Phi(Y^("an "))\Phi\left(Y^{\text {an }}\right). The following terminology will be convenient:
Definition 3.19. Let SS be a smooth quasiprojective complex variety. Let V\mathbb{V} be a polarized Z\mathbb{Z} VHS on S^("an ")S^{\text {an }} with period map Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash D. A closed irreducible subvariety Y sub SY \subset S is called a special subvariety of SS for V\mathbb{V} if it coincides with an irreducible component of the preimage Phi^(-1)(Gamma_(Y)\\D_(Y))\Phi^{-1}\left(\Gamma_{Y} \backslash D_{Y}\right).
Equivalently, a special subvariety of SS for V\mathbb{V} is a closed irreducible algebraic subvariety Y sub SY \subset S maximal among the closed irreducible algebraic subvarieties ZZ of SS such that the generic Mumford-Tate group G_(Z)\mathbf{G}_{Z} of V_(∣Z)\mathbb{V}_{\mid Z} equals G_(Y)\mathbf{G}_{Y}.
3.6.2. A conjecture of Griffiths
Combining Theorem 3.14 this time with Theorem 3.12 leads to a proof of an old conjecture of Griffiths [44], claiming that the image of any period map has a natural structure of quasiprojective variety (Griffiths proved it when the target Hodge variety is compact):
Theorem 3.20 ([10]). Let SS be a smooth connected quasiprojective complex variety and let Phi:S^(an)rarr Gamma\\D\Phi: S^{\mathrm{an}} \rightarrow \Gamma \backslash D be a period map. There exists a unique dominant morphism of complex algebraic varieties f:S rarr Tf: S \rightarrow T, with TT quasiprojective, and a closed complex analytic immersion iota:T^(an)↪Gamma\\D\iota: T^{\mathrm{an}} \hookrightarrow \Gamma \backslash D such that Phi=iota@f^(an)\Phi=\iota \circ f^{\mathrm{an}}.
Let us sketch the proof. As before, let S sub bar(S)S \subset \bar{S} be a log-smooth compactification by a simple normal crossing divisor ZZ. It follows from a result of Griffiths [43, PROP. 9.11I)] that Phi\Phi extends to a proper period map over the components of ZZ around which the monodromy is finite. Hence, without loss of generality, we can assume that Phi\Phi is proper. The existence of ff in the category of algebraic spaces then follows immediately from Theorems 3.14 and 3.12 (for S=R_(an",exp ")S=\mathbb{R}_{\mathrm{an} \text {,exp }} ). The proof that TT is in fact quasiprojective exploits a crucial observation of Griffiths that Gamma\\D\Gamma \backslash D carries a positively curved Q\mathbb{Q}-line bundle L:=ox_(p)det(F^(p))\mathscr{L}:=\otimes_{p} \operatorname{det}\left(F^{p}\right). This line bundle is naturally definable on (Gamma\\D)^("def ")(\Gamma \backslash D)^{\text {def }}. Using the definable GAGA Theorem 3.11, one shows that its restriction to T^("def ")T^{\text {def }} comes from an algebraic Q\mathbb{Q}-line bundle L_(T)L_{T} on TT, which one manages to show to be ample.
4. FUNCTIONAL TRANSCENDENCE
4.1. Bialgebraic geometry
As we saw, Hodge theory, which compares the Hodge filtration on H_(dR)^(∙)(X//C)H_{\mathrm{dR}}^{\bullet}(X / \mathbb{C}) with the rational structure on H_(B)^(∙)(X^(an),C)H_{\mathrm{B}}^{\bullet}\left(X^{\mathrm{an}}, \mathbb{C}\right), gives rise to variational Hodge theory, whose fundamental diagram (2.2) compares the algebraic structure of SS with the algebraic structure on the dual period domain D^(ˇ)\check{D}. As such, it is a partial answer to one of the most classical problem of complex algebraic geometry: the transcendental nature of the topological universal cover of complex algebraic varieties. If SS is a connected complex algebraic variety, the universal cover widetilde(S^(an))\widetilde{S^{\mathrm{an}}} has usually no algebraic structure as soon as the topological fundamental group pi_(1)(S^(an))\pi_{1}\left(S^{\mathrm{an}}\right) is infinite. As an aside, let us mention an interesting conjecture of Kóllar and Pardon [60], predicting that if XX is a normal projective irreducible complex variety whose universal cover widetilde(X^("an "))\widetilde{X^{\text {an }}} is biholomorphic to a semialgebraic open subset of an algebraic variety then widetilde(X^("an "))\widetilde{X^{\text {an }}} is biholomorphic to C^(n)xx D xxF^("an ")\mathbb{C}^{n} \times D \times F^{\text {an }}, where DD is a bounded symmetric domain and FF is a normal, projective, irreducible, topologically simply connected, complex algebraic variety We want to think of variational Hodge theory as an attempt to provide a partial algebraic uniformization: the period map emulates an algebraic structure on widetilde(S^(an))\widetilde{S^{\mathrm{an}}}, modeled on the flag variety D^(ˇ)\check{D}. The remaining task is then to describe the transcendence properties of the complex analytic uniformization map p: widetilde(S^(an))rarrS^(an)p: \widetilde{S^{\mathrm{an}}} \rightarrow S^{\mathrm{an}} with respects to the emulated algebraic structure on widetilde(S^(an))\widetilde{S^{\mathrm{an}}} and the algebraic structure SS on S^("an ")S^{\text {an }}. A few years ago, the author [55], together with Ullmo and Yafaev [59], introduced a convenient format for studying such questions, which encompasses many classical transcendence problems and provides a powerful heuristic.
Definition 4.1. A bialgebraic structure on a connected quasiprojective variety SS over C\mathbb{C} is a pair
where ZZ denotes an algebraic variety (called the algebraic model of widetilde(S^(an))\widetilde{S^{\mathrm{an}}} ), Aut(Z)\operatorname{Aut}(Z) is its group of algebraic automorphisms, rho\rho is a group morphism (called the monodromy representation) and ff is a rho\rho-equivariant holomorphic map (called the developing map).
An irreducible analytic subvariety Y sub widetilde(S^(an))Y \subset \widetilde{S^{\mathrm{an}}} is said to be an algebraic subvariety of widetilde(S^("an "))\widetilde{S^{\text {an }}} for the bialgebraic structure (f,rho)(f, \rho) if YY is an analytic irreducible component of f^(-1)( bar(f(Y))^("Zar "):}f^{-1}\left(\overline{f(Y)}^{\text {Zar }}\right. ) (where bar(f(Y))^("Zar ")\overline{f(Y)}^{\text {Zar }} denotes the Zariski-closure of f(Y)f(Y) in ZZ ). An irreducible algebraic subvariety Y sub widetilde(S^(an))Y \subset \widetilde{S^{\mathrm{an}}}, resp. W sub SW \subset S, is said to be bialgebraic if p(Y)p(Y) is an algebraic subvariety of SS, resp. any (equivalently one) analytic irreducible component of p^(-1)(W)p^{-1}(W) is an irreducible algebraic subvariety of widetilde(S^(an))\widetilde{S^{\mathrm{an}}}. The bialgebraic subvarieties of SS are precisely the ones where the emulated algebraic structure on widetilde(S^(an))\widetilde{S^{\mathrm{an}}} and the one on SS interact nontrivially.
Example 4.2. (a) tori, S=(C^(**))^(n)S=\left(\mathbb{C}^{*}\right)^{n}. The uniformization map is the multiexponential
p:=(exp(2pi i*),dots,exp(2pi i*)):C^(n)rarr(C^(**))^(n)p:=(\exp (2 \pi i \cdot), \ldots, \exp (2 \pi i \cdot)): \mathbb{C}^{n} \rightarrow\left(\mathbb{C}^{*}\right)^{n}
and ff is the identity morphism of C^(n)\mathbb{C}^{n}. An irreducible algebraic subvariety Y subC^(n)Y \subset \mathbb{C}^{n} (resp. {:W sub(C^(**))^(n))\left.W \subset\left(\mathbb{C}^{*}\right)^{n}\right) is bialgebraic if and only if YY is a translate of a rational linear subspace of C^(n)=Q^(n)ox_(Q)C\mathbb{C}^{n}=\mathbb{Q}^{n} \otimes_{\mathbb{Q}} \mathbb{C} (resp. WW is a translate of a subtorus of {:(C^(**))^(n))\left.\left(\mathbb{C}^{*}\right)^{n}\right)
(b) abelian varieties, S=AS=A is a complex abelian variety of dimension nn. Let p:p: Lie A≃A \simeqC^(n)rarr A\mathbb{C}^{n} \rightarrow A be the uniformizing map of a complex abelian variety AA of dimension nn. Once more widetilde(S^(an))=C^(n)\widetilde{S^{\mathrm{an}}}=\mathbb{C}^{n} and ff is the identity morphism. One checks easily that an irreducible algebraic subvariety W sub AW \subset A is bialgebraic if and only if WW is the translate of an abelian subvariety of AA.
(c) Shimura varieties, (G,D)(\mathbf{G}, D) is a Shimura datum. The quotient S^("an ")=Gamma\\DS^{\text {an }}=\Gamma \backslash D (for Gamma sub G:=\Gamma \subset G:=G^("der ")(R)^(+)\mathbf{G}^{\text {der }}(\mathbb{R})^{+}a congruence torsion-free lattice) is the complex analytification of a (connected) Shimura variety Sh, defined over a number field (a finite extension of the reflex field of (G,D)(\mathbf{G}, D) ). And ff is the open embedding D↪D^("an ")D \hookrightarrow D^{\text {an }}.
Let us come back to the case of the bialgebraic structure on SS
defined by a polarized ZVHSV\mathbb{Z V H S} \mathbb{V} and its period map Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash D with monodromy rho\rho : pi_(1)(S^(an))rarr Gamma subG(Q)\pi_{1}\left(S^{\mathrm{an}}\right) \rightarrow \Gamma \subset \mathbf{G}(\mathbb{Q}) (in fact, all the examples above are of this form if we consider more generally graded-polarized variations of mixed Z\mathbb{Z}-Hodge structures). What are its bialgebraic subvarieties? To answer this question, we need to define the weakly special subvarieties of Gamma\\D\Gamma \backslash D, as either a special subvariety or a subvariety of the form
where (HxxL,D_(H)xxD_(L))\left(\mathbf{H} \times \mathbf{L}, D_{H} \times D_{L}\right) is a Hodge subdatum of (G^("ad "),D)\left(\mathbf{G}^{\text {ad }}, D\right) and {t}\{t\} is a Hodge generic point in Gamma_(L)\\D_(L)\Gamma_{\mathbf{L}} \backslash D_{L}. Generalizing Theorem 3.17, the preimage under Phi\Phi of any weakly special subvariety of Gamma\\D\Gamma \backslash D is an algebraic subvariety of SS [56]. An irreducible component of such a preimage is called a weakly special subvariety of SS for V\mathbb{V} (or Phi\Phi ).
Theorem 4.3 ([56]). Let Phi:S^(an)rarr Gamma\\D\Phi: S^{\mathrm{an}} \rightarrow \Gamma \backslash D be a period map. The bialgebraic subvarieties of SS for the bialgebraic structure defined by Phi\Phi are precisely the weakly special subvarieties of SS for Phi\Phi. In analogy with Definition 3.19, they are also the closed irreducible algebraic subvarieties Y sub SY \subset S maximal among the closed irreducible algebraic subvarieties ZZ of SS whose algebraic monodromy group H_(Z)\mathbf{H}_{Z} equals H_(Y)\mathbf{H}_{Y}.
When S=ShS=\mathrm{Sh} is a Shimura variety, these results are due to Moonen [65] and [91]. In that case the weakly special subvarieties are also the irreducible algebraic subvarieties of Sh whose smooth locus is totally geodesic in Sh^("an ")\mathrm{Sh}^{\text {an }} for the canonical Kähler-Einstein metric on Sh^("an ")=Gamma\\D\mathrm{Sh}^{\text {an }}=\Gamma \backslash D coming from the Bergman metric on DD, see [65].
To study not only functional transcendence but also arithmetic transcendence, we enrich bialgebraic structures over bar(Q)\overline{\mathbb{Q}}. A bar(Q)\overline{\mathbb{Q}}-bialgebraic structure on a quasi-projective variety SS defined over bar(Q)\overline{\mathbb{Q}} is a bialgebraic structure (f:( widetilde(S^(an)))rarrZ^(an),h:pi_(1)(S^("an "))rarr Aut(Z))\left(f: \widetilde{S^{\mathrm{an}}} \rightarrow Z^{\mathrm{an}}, h: \pi_{1}\left(S^{\text {an }}\right) \rightarrow \operatorname{Aut}(Z)\right) such that ZZ is defined over bar(Q)\overline{\mathbb{Q}} and the homomorphism hh takes values in Aut bar(Q)Z\overline{\mathbb{Q}} Z. An algebraic subvariety Y sub widetilde(S^("an "))Y \subset \widetilde{S^{\text {an }}} is said to be defined over bar(Q)\overline{\mathbb{Q}} if its model bar(f(Y))^("Zar ")sub Z\overline{f(Y)}^{\text {Zar }} \subset Z is. A bar(Q)\overline{\mathbb{Q}}-bialgebraic subvariety W sub SW \subset S is an algebraic subvariety of SS defined over bar(Q)\overline{\mathbb{Q}} and such that any (equivalently one) of the analytic irreducible components of p^(-1)(W)p^{-1}(W) is an algebraic subvariety of widetilde(S^(an))\widetilde{S^{\mathrm{an}}} defined over bar(Q)\overline{\mathbb{Q}}. A bar(Q)\overline{\mathbb{Q}}-bialgebraic point s in S( bar(Q))s \in S(\overline{\mathbb{Q}}) is also called an arithmetic point. Example 4.2a)4.2 \mathrm{a}) is naturally defined over bar(Q)\overline{\mathbb{Q}}, with arithmetic points the torsion points of (C^(**))^(n)\left(\mathbb{C}^{*}\right)^{n}. In Example 4.2 b) the bialgebraic structure can be defined over bar(Q)\overline{\mathbb{Q}} if the abelian variety AA has CM\mathrm{CM}, and its arithmetic points are its torsion points, see [90]. Example 4.2c) is naturally a bar(Q)\overline{\mathbb{Q}}-bialgebraic structure, with arithmetic points the special points of the Shimura variety (namely the special subvarieties of dimension zero), at least when the pure part of the Shimura variety is of Abelian type, see [84]. In all these cases it is interesting to notice that the bar(Q)\overline{\mathbb{Q}}-bialgebraic subvarieties are the bialgebraic subvarieties containing one arithmetic point (in Example 4.2c) these are the special subvarieties of the Shimura variety).
The bi-algebraic structure associated with a period map Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash D is defined over bar(Q)\overline{\mathbb{Q}} as soon as SS is. In this case, we expect the bar(Q)\overline{\mathbb{Q}}-bi-algebraic subvarieties to be precisely the special subvarieties, see [55, 2.6 AND 3.4].
4.2. The Ax-Schanuel theorem for period maps
The geometry of bialgebraic structures is controlled by the following functional transcendence heuristic, whose idea was introduced by Pila in the case of Shimura varieties, see [73,74]:[73,74]:
Ax-Schanuel principle. Let SS be an irreducible algebraic variety endowed with a non-trivial bialgebraic structure. Let U sub widetilde(S^("an "))xxS^("an ")U \subset \widetilde{S^{\text {an }}} \times S^{\text {an }} be an algebraic subvariety (for the product bialgebraic structure) and let WW be an analytic irreducible component of U nn DeltaU \cap \Delta, where Delta\Delta denotes the graph of p: widetilde(S^(an))rarrS^(an)p: \widetilde{S^{\mathrm{an}}} \rightarrow S^{\mathrm{an}}. Then codim_(U)W >= dim bar(W)^(bi)\operatorname{codim}_{U} W \geq \operatorname{dim} \bar{W}^{\mathrm{bi}}, where bar(W)^(bi)\bar{W}^{\mathrm{bi}} denotes the smallest bialgebraic subvariety of SS containing p(W)p(W).
When applied to a subvariety U sub widetilde(S^(an))xxS^("an ")U \subset \widetilde{S^{\mathrm{an}}} \times S^{\text {an }} of the form Y xx bar(p(Y))^("Zar ")Y \times \overline{p(Y)}^{\text {Zar }} for Y sub widetilde(S^(an))Y \subset \widetilde{S^{\mathrm{an}}} algebraic, the Ax-Schanuel principle specializes to the following:
Ax-Lindemann principle. Let SS be an irreducible algebraic variety endowed with a nontrivial bialgebraic structure. Let Y sub widetilde(S^(an))Y \subset \widetilde{S^{\mathrm{an}}} be an algebraic subvariety. Then bar(p(Y))^(Zar)\overline{p(Y)}^{\mathrm{Zar}} is a bialgebraic subvariety of SS.
Ax [5,6][5,6] showed that the abstract Ax-Schanuel principle holds true for Example 4.2a) and Example 4.2b) above, using differential algebra. Notice that the Ax-Lindemann principle in Example 4.2a) is the functional analog of the classical Lindemann theorem stating that if alpha_(1),dots,alpha_(n)\alpha_{1}, \ldots, \alpha_{n} are Q\mathbb{Q}-linearly independent algebraic numbers then e^(alpha_(1)),dots,e^(alpha_(n))e^{\alpha_{1}}, \ldots, e^{\alpha_{n}} are algebraically independent over Q\mathbb{Q}. This explains the terminology. The Ax-Lindemann principle in Example 4.2c) was proven by Pila [72] when SS is a product Y(1)^(n)xx(C^(**))^(k)Y(1)^{n} \times\left(\mathbb{C}^{*}\right)^{k}, by Ullmo-Yafaev [92] for projective Shimura varieties, by Pila-Tsimerman [76] for A_(g)\mathscr{A}_{g}, and by Klingler-Ullmo-Yafaev [58] for any pure Shimura variety. The full Ax-Schanuel principle was proven by Mok-Pila-Tsimerman for pure Shimura varieties [64].
We conjectured in [55, coNJ. 7.5] that the Ax-Schanuel principle holds true for the bi-algebraic structure associated to a (graded-)polarized variation of (mixed) ZHS\mathbb{Z H S} on an arbitrary quasiprojective variety SS. Bakker and Tsimerman proved this conjecture in the pure case:
Theorem 4.4 (Ax-Schanuel for ZVHS,[12]\mathbb{Z V H S ,}[12] ). Let Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash D be a period map. Let V sub S xxD^(ˇ)V \subset S \times \check{D} be an algebraic subvariety. Let UU be an irreducible complex analytic component of W nn(Sxx_(Gamma\\D)D)W \cap\left(S \times_{\Gamma \backslash D} D\right) such that
Then the projection of UU to SS is contained in a strict weakly special subvariety of SS for Phi\Phi.
Remark 4.5. The results of [64] were extended by Gao [39] to mixed Shimura varieties of Kuga type. Recently the full Ax-Schanuel [55, coNJ. 7.5] for variations of mixed Hodge structures has been fully proven independently in [40] and [26].
The proof of Theorem 4.4 follows a strategy started in [58] and fully developed in [64] in the Shimura case, see [88] for an introduction. It does not use Theorem 3.14, but only a weak version equivalent to the Nilpotent Orbit Theorem, and relies crucially on the definable Chow Theorem 3.10, the Pila-Wilkie Theorem 3.8, and the proof that the volume (for the natural metric on Gamma\\D)\Gamma \backslash D) of the intersection of a ball of radius RR in Gamma\\D\Gamma \backslash D with the horizontal complex analytic subvariety Phi(S^("an "))\Phi\left(S^{\text {an }}\right) grows exponentially with RR (a negative curvature property of the horizontal tangent bundle).
Using the Ax-Lindemann theorem special case of Theorem 4.4 and a global algebraicity result in the total bundle of V\mathcal{V}, Otwinowska and the author proved the following:
Theorem 4.6 ([56]). Let V\mathbb{V} be a polarized ZVHS\mathbb{Z} V H S on a smooth connected complex quasiprojective variety SS. Then either HL (S,V^(ox))_(fpos)\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{fpos}} is Zariski-dense in SS; or it is an algebraic subvariety of SS (i.e., the set of strict special subvarieties of SS for V\mathbb{V} of factorwise positive period dimension has only finitely many maximal elements for the inclusion).
Example 4.7. The simplest example of Theorem 4.6 is the following. Let S subA_(g)S \subset \mathscr{A}_{g} be a Hodge-generic closed irreducible subvariety. Either the set of positive-dimensional closed
irreducible subvarieties of SS which are not Hodge generic has finitely many maximal elements (for the inclusion), or their union is Zariski-dense in SS.
Example 4.8. Let B subPH^(0)(P_(C)^(3),O(d))B \subset \mathbb{P} H^{0}\left(\mathbb{P}_{\mathbb{C}}^{3}, \mathcal{O}(d)\right) be the open subvariety parametrizing the smooth surfaces of degree dd in P_(C)^(3)\mathbb{P}_{\mathbb{C}}^{3}. Suppose d > 3d>3. The classical Noether theorem states that any surface Y subP_(C)^(3)Y \subset \mathbb{P}_{\mathbb{C}}^{3} corresponding to a very general point [Y]in B[Y] \in B has Picard group Z\mathbb{Z} : every curve on YY is a complete intersection of YY with another surface in P_(C)^(3)\mathbb{P}_{\mathbb{C}}^{3}. The countable union NL(B)\mathrm{NL}(B) of closed algebraic subvarieties of BB corresponding to surfaces with bigger Picard group is called the Noether-Lefschetz locus of BB. Let Vrarr B\mathbb{V} \rightarrow B be the ZVHSR^(2)f_(**)Z_("prim ")\mathbb{Z V H S} R^{2} f_{*} \mathbb{Z}_{\text {prim }}, where f:y rarr Bf: y \rightarrow B denotes the universal family of surfaces of degree dd. Clearly NL(B)sub\mathrm{NL}(B) \subsetHL(B,V^(ox))\operatorname{HL}\left(B, \mathbb{V}^{\otimes}\right). Green (see [94, PRoP. 5.20]) proved that NL(B)\operatorname{NL}(B), hence also HL(B,V^(ox))\operatorname{HL}\left(B, \mathbb{V}^{\otimes}\right), is analytically dense in BB. Now Theorem 4.6 implies the following: Let S sub BS \subset B be a Hodgegeneric closed irreducible subvariety. Either S nn HL (B,V^(ox))_("fpos ")S \cap \operatorname{HL}\left(B, \mathbb{V}^{\otimes}\right)_{\text {fpos }} contains only finitely many maximal positive-dimensional closed irreducible subvarieties of SS, or the union of such subvarieties is Zariski-dense in SS.
5. TYPICAL AND ATYPICAL INTERSECTIONS: THE ZILBER-PINK CONJECTURE FOR PERIOD MAPS
5.1. The Zilber-Pink conjecture for ZVHS\mathbb{Z V H S} : Conjectures
In the same way that the Ax-Schanuel principle controls the geometry of bialgebraic structures, the diophantine geometry of bar(Q)\overline{\mathbb{Q}}-bialgebraic structures is controlled by the following heuristic:
Atypical intersection principle. Let SS be an irreducible algebraic bar(Q)\overline{\mathbb{Q}}-variety endowed with a bar(Q)a \overline{\mathbb{Q}}-bialgebraic structure. Then the union S_("atyp ")S_{\text {atyp }} of atypical bar(Q)\overline{\mathbb{Q}}-bialgebraic subvarieties of SS is an algebraic subvariety of SS (i.e., it contains only finitely many atypical bar(Q)\overline{\mathbb{Q}}-bialgebraic subvarieties maximal for the inclusion).
Here a bar(Q)\overline{\mathbb{Q}}-bialgebraic subvariety Y sub SY \subset S is said to be atypical for the given bialgebraic structure on SS if it is obtained as an excess intersection of f(( widetilde(S^("an "))))f\left(\widetilde{S^{\text {an }}}\right) with its model bar(f(( tilde(Y))))\overline{f(\tilde{Y})} Zar sub Z\subset Z; and S_("atyp ")S_{\text {atyp }} denotes the union of all atypical subvarieties of SS. As a particular case of the atypical intersection principle:
Sparsity of arithmetic points principle. Let SS be an irreducible algebraic bar(Q)\overline{\mathbb{Q}}-variety endowed with a bar(Q)\overline{\mathbb{Q}}-bialgebraic structure. Then any irreducible algebraic subvariety of SS containing a Zariski-dense set of atypical arithmetic points is a bar(Q)\overline{\mathbb{Q}}-bialgebraic subvariety.
For a general polarized ZVHSV\mathbb{Z V H S} \mathbb{V} with period map Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash D, which we can assume to be proper without loss of generality, we already mentioned that even the geometric characterization of the bar(Q)\overline{\mathbb{Q}}-bialgebraic subvarieties as the special subvarieties is unknown. Replacing the bar(Q)\overline{\mathbb{Q}}-bialgebraic subvarieties of SS by the special ones, we define:
Definition 5.1. A special subvariety Z=Phi^(-1)(Gamma_(Z)\\D_(Z))^(0)sub SZ=\Phi^{-1}\left(\Gamma_{Z} \backslash D_{Z}\right)^{0} \subset S is said atypical if either ZZ is singular for V\mathbb{V} (meaning that Phi(Z^(an))\Phi\left(Z^{\mathrm{an}}\right) is contained in the singular locus of the complex analytic variety Phi(S^(an))\Phi\left(S^{\mathrm{an}}\right) ), or if Phi(S^(an))\Phi\left(S^{\mathrm{an}}\right) and Gamma_(Z)\\D_(Z)\Gamma_{Z} \backslash D_{Z} do not intersect generically along Phi(Z)\Phi(Z) :
Defining the atypical Hodge locus HL(S,V^(ox))_("atyp ")subHL(S,V^(ox))\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\text {atyp }} \subset \mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right) as the union of the atypical special subvarieties of SS for V\mathbb{V}, we obtain the following precise atypical intersection principle for ZVHS\mathbb{Z V H S}, first proposed in [55] in a more restrictive form:
Conjecture 5.2 (Zilber-Pink conjecture for ZVHS,[13,55])\mathbb{Z V H S},[13,55]). Let V\mathbb{V} be a polarizable ZVHS\mathbb{Z} V H S on an irreducible smooth quasiprojective variety SS. The atypical Hodge locus HL(S,V^(ox))_("atyp ")\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\text {atyp }} is a finite union of atypical special subvarieties of SS for V\mathbb{V}. Equivalently, the set of atypical special subvarieties of SS for V\mathbb{V} has finitely many maximal elements for the inclusion.
Notice that this conjecture is in some sense more general than the above atypical intersection principle, as we do not assume that SS is defined over bar(Q)\overline{\mathbb{Q}}; this has to be compared to the fact that the Manin-Mumford conjecture holds true for every complex abelian variety, not necessarily defined over bar(Q)\overline{\mathbb{Q}}.
Example 5.3. Recently Baldi and Ullmo [14] proved a special case of Conjecture 5.2 of much interest. Margulis' arithmeticity theorem states that any lattice in a simple real Lie group GG of real rank at least 2 is arithmetic: it is commensurable with a group G(Z)\mathbf{G}(\mathbb{Z}), for G\mathbf{G} a Q\mathbb{Q} algebraic group such that G(R)=G\mathbf{G}(\mathbb{R})=G up to a compact factor. On the other hand, the structure of lattices in a simple real Lie group of rank 1, like the group PU(n,1)\operatorname{PU}(n, 1) of holomorphic isometries of the complex unit ball B_(C)^(n)\mathbf{B}_{\mathbb{C}}^{n} endowed with its Bergman metric, is an essentially open question. In particular, there exist nonarithmetic lattices in PU(n,1),n=2,3\operatorname{PU}(n, 1), n=2,3. Let iota:Lambda↪PU(n,1)\iota: \Lambda \hookrightarrow \operatorname{PU}(n, 1) be a lattice. The ball quotient S^(an):=Lambda\\B_(C)^(n)S^{\mathrm{an}}:=\Lambda \backslash \mathbf{B}_{\mathbb{C}}^{n} is the analytification of a complex algebraic variety SS. By results of Simpson and Esnault-Groechenig, there exists a ZVHSPhi:S^("an ")rarr Gamma\\(B_(C)^(n)xxD^('))\mathbb{Z V H S} \Phi: S^{\text {an }} \rightarrow \Gamma \backslash\left(\mathbf{B}_{\mathbb{C}}^{n} \times D^{\prime}\right) with monodromy representation rho:Lambda rarrPU(n,1)xxG^(')\rho: \Lambda \rightarrow \mathrm{PU}(n, 1) \times G^{\prime}
whose first factor Lambda rarrPU(n,1)\Lambda \rightarrow \mathrm{PU}(n, 1) is the rigid representation iota\iota. The special subvarieties of SS for V\mathbb{V} are the totally geodesic complex subvarieties of S^("an ")S^{\text {an }}. When Lambda\Lambda is nonarithmetic, they are automatically atypical. In accordance with Conjecture 5.2 in this case, Baldi and Ullmo prove that if Lambda\Lambda is nonarithmetic, then S^("an ")S^{\text {an }} contains only finitely many maximal totally geodesic subvarieties. This result has been proved independently by Bader, Fisher, Miller, and Stover [7], using completely different methods from homogeneous dynamics.
Among the special points for a ZVHSV\mathbb{Z V H S} \mathbb{V}, the CM\mathrm{CM}-points (i.e., those for which the Mumford-Tate group is a torus) are always atypical except if the generic Hodge datum (G,D)(\mathbf{G}, D) is of Shimura type and the period map Phi\Phi is dominant. Hence, as explained in [55, SECTION 5.2], Conjecture 5.2 implies the following:
Example 5.5. Consider the Calabi-Yau Hodge structure VV of weight 3 with Hodge numbers h^(3,0)=h^(2,1)=1h^{3,0}=h^{2,1}=1 given by the mirror dual quintic. Its universal deformation space SS is the projective line minus 3 points, which carries a ZVHSV\mathbb{Z V H S} \mathbb{V} of the same type. This gives a nontrivial period map Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash D, where D=Sp(4,R)//U(1)xx U(1)D=\mathbf{S p}(4, \mathbb{R}) / U(1) \times U(1) is a 4-dimensional period domain. This period map is known not to factorize through a Shimura subvariety (its algebraic monodromy group is Sp4\mathbf{S p} 4 ). Conjecture 5.4 in that case predicts that SS contains only finitely many points CM-points ss. A version of this prediction already appears in [48]. The more general Conjecture 5.2 also predicts that SS contains only finitely many points ss where V_(s)\mathbb{V}_{s} splits as a direct sum of two (Tate twisted) weight one Hodge structures (V_(s)^(2,1)o+V_(s)^(1,2))\left(\mathbb{V}_{s}^{2,1} \oplus \mathbb{V}_{s}^{1,2}\right) and its orthogonal for the Hodge metric (V_(s)^(3,0)o+V_(s)^(0,3))\left(\mathbb{V}_{s}^{3,0} \oplus \mathbb{V}_{s}^{0,3}\right) (the so-called "rank two attractors" points, see [66]).
Conjecture 5.2 about the atypical Hodge locus takes all its meaning if we compare it to the expected behavior of its complement, the typical Hodge locus HL (S,V^(ox))_(typ):=\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{typ}}:=HL(S,V^(ox))\\HL(S,V_("atyp ")^(ox)):\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right) \backslash \mathrm{HL}\left(S, \mathbb{V}_{\text {atyp }}^{\otimes}\right):
Conjecture 5.6 (Density of the typical Hodge locus, [13]). If HL(S,V^(ox))_(typ)\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{typ}} is not empty then it is dense (for the analytic topology) in S^(an)S^{\mathrm{an}}.
Conjectures 5.2 and 5.6 imply immediately the following, which clarifies the possible alternatives in Theorem 4.6:
Conjecture 5.7 ([13]). Let V\mathbb{V} be a polarizable ZVHS\mathbb{Z} V H S on an irreducible smooth quasiprojective variety SS. If HL(S,V^(ox))_(typ)\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{typ}} is empty then HL(S,V^(ox))\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right) is algebraic; otherwise, HL(S,V^(ox))\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right) is analytically dense in S^("an ")S^{\text {an }}.
5.2. The Zilber-Pink conjecture for Z\mathbb{Z} VHS: Results
In [13] Baldi, Ullmo, and I establish the geometric part of Conjecture 5.2: the maximal atypical special subvarieties of positive period dimension arise in a finite number of families whose geometry is well understood. We cannot say anything on the atypical locus of zero period dimension (for which different ideas are certainly needed):
Theorem 5.8 (Geometric Zilber-Pink, [13]). Let V\mathbb{V} be a polarizable ZVHS\mathbb{Z} V H S on a smooth connected complex quasiprojective variety SS. Let ZZ be an irreducible component of the Zariski closure of HL(S,V^(ox))_(pos," atyp "):=HL(S,V^(ox))_(pos)nnHL(S,V^(ox))_("atyp ")\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{pos}, \text { atyp }}:=\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{pos}} \cap \mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\text {atyp }} in SS. Then:
(a) Either ZZ is a maximal atypical special subvariety;
such that Z contains a Zariski-dense set of atypical special subvarieties for Phi^('')\Phi^{\prime \prime} of zero period dimension. Moreover, ZZ is Hodge generic in the special subvariety Phi^(-1)(Gamma_(G_(Z))\\D_(G_(Z)))^(0)\Phi^{-1}\left(\Gamma_{\mathbf{G}_{Z}} \backslash D_{G_{Z}}\right)^{0} of SS for Phi\Phi, which is typical.
Conjecture 5.2, which also takes into account the atypical special subvarieties of zero period dimension, predicts that the branch (b) of the alternative in the conclusion of Theorem 5.8 never occurs. Theorem 5.8 is proven using properties of definable sets and the Ax-Schanuel Theorem 4.4, following an idea originating in [89].
Theorem 5.9 ([13]). Let V\mathbb{V} be a polarizable ZVHS\mathbb{Z} V H S on a smooth irreducible complex quasiprojective variety SS, with generic Hodge datum (G,D)(\mathbf{G}, D). Suppose that the Shimura locus of SS for V\mathbb{V} of positive period dimension is Zariski-dense in SS. If G^("ad ")\mathbf{G}^{\text {ad }} is simple then V\mathbb{V} is of Shimura type.
5.3. On the algebraicity of the Hodge locus
In view of Conjecture 5.7, it is natural to ask if there a simple combinatorial criterion on (G,D)(\mathbf{G}, D) for deciding whether HL(S,V)_("typ ")\operatorname{HL}(S, \mathbb{V})_{\text {typ }} is empty. Intuitively, one expects that the more "complicated" the Hodge structure is, the smaller the typical Hodge locus should be, due to the constraint imposed by Griffiths' transversality. Let us measure the complexity of
V\mathbb{V} by its level: when G^("ad ")\mathbf{G}^{\text {ad }} is simple, it is the greatest integer kk such that g^(k,-k)!=0\mathrm{g}^{k,-k} \neq 0 in the Hodge decomposition of the Lie algebra gg of G\mathbf{G}; in general one takes the minimum of these integers obtained for each simple Q\mathbb{Q}-factor of G^("ad ")\mathbf{G}^{\text {ad }}. While strict typical special subvarieties
usually abound for ZVHSs\mathbb{Z V H S s} of level one (e.g., families of abelian varieties, see Example 4.7; or families of K3 surfaces) and can occur in level two (see Example 4.8), they do not exist in level at least three!
Theorem 5.10 ([13]). Let V\mathbb{V} be a polarizable ZVHS\mathbb{Z} V H S on a smooth connected complex quasiprojective variety SS. If V\mathbb{V} is of level at least 3 then HL (S,V^(ox))_(typ)=O/\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{typ}}=\emptyset (and thus HL(S,V^(ox))=\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)={: HL (S,V^(ox))_("atyp "))\left.\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\text {atyp }}\right).
The proof of Theorem 5.10 is purely Lie-theoretic. Let (G,D)(\mathbf{G}, D) be the generic Hodge datum of V\mathbb{V} and Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash D its period map. Suppose that Y sub SY \subset S is a typical special subvariety, with generic Hodge datum ( G_(Y),D_(Y)\mathbf{G}_{Y}, D_{Y} ). The typicality condition and the horizontality of the period map Phi\Phi imply that g_(Y)^(-i,i)=g^(-i,i)\mathrm{g}_{Y}^{-i, i}=\mathrm{g}^{-i, i} for all i >= 2i \geq 2 (for the Hodge structures on the Lie algebras g_(Y)\mathrm{g}_{Y} and g\mathrm{g} defined by some point of D_(Y)D_{Y} ). Under the assumption that V\mathbb{V} has level at least 3, we show that this is enough to ensure that g_(Y)=g\mathrm{g}_{Y}=\mathrm{g}, hence Y=SY=S. Hence there are no strict typical special subvariety.
Notice that Conjecture 5.2 and Theorem 5.10 imply:
Conjecture 5.11 (Algebraicity of the Hodge locus in level at least 3, [13]). Let V\mathbb{V} be a polarizable ZVHS\mathbb{Z} V H S on a smooth connected complex quasiprojective variety SS. If V\mathbb{V} is of level at least 3 then HL(S,V^(ox))\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right) is algebraic.
The main result of [13], which follows immediately from Theorems 5.8 and 5.10, is the following stunning geometric reinforcement of Theorems 3.1 and 4.6:
Theorem 5.12 ([13]). If V\mathbb{V} is of level at least 3 then HL (S,V^(ox))_(fpos)\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{fpos}} is algebraic.
As a simple geometric illustration of Theorem 5.12, we prove the following, to be contrasted with the n=2n=2 case (see Example 4.8):
Corollary 5.13. Let P_(C)^(N(n,d))\mathbf{P}_{\mathbb{C}}^{N(n, d)} be the projective space parametrizing the hypersurfaces XX of P_(C)^(n+1)\mathbf{P}_{\mathbb{C}}^{n+1} of degree d (where N(n,d)=((n+d+1)/(d))-1N(n, d)=\binom{n+d+1}{d}-1 ). Let U_(n,d)subP_(C)^(N(n,d))U_{n, d} \subset \mathbf{P}_{\mathbb{C}}^{N(n, d)} be the Zariskiopen subset parametrizing the smooth hypersurfaces XX and let VrarrU_(n,d)\mathbb{V} \rightarrow U_{n, d} be the ZVHS\mathbb{Z} V H S corresponding to the primitive cohomology H^(n)(X,Z)_("prim. ")H^{n}(X, \mathbb{Z})_{\text {prim. }}. If n >= 3n \geq 3 and d > 5d>5, then HL (U_(n,d),V^(ox))_(pos)subU_(n,d)\operatorname{HL}\left(U_{n, d}, \mathbb{V}^{\otimes}\right)_{\mathrm{pos}} \subset U_{n, d} is algebraic.
5.4. On the typical Hodge locus in level one and two
In the direction of Conjecture 5.6, we obtain:
Theorem 5.14 (Density of the typical locus, [13]). Let V\mathbb{V} be a polarized ZVHS\mathbb{Z} V H S on a smooth connected complex quasiprojective variety S. If the typical Hodge locus HL(S,V^(ox))_(typ)\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{typ}} is nonempty (hence the level of V\mathbb{V} is one or two by Theorem 5.10) then HL(S,V^(ox))\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right) is analytically (hence Zariski) dense in SS.
Notice that, in Theorem 5.14, we also treat the typical Hodge locus of zero period dimension. Theorem 5.14 is new even for SS a subvariety of a Shimura variety. Its proof is inspired by the arguments of Chai [24] in that case.
It remains to find a criterion for deciding whether, in level one or two, the typical Hodge locus HL (S,V^(ox))_("typ ")\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\text {typ }} is empty or not. We refer to [57, THEOREM 2.15] and [85, 86] for results in this direction.
6. ARITHMETIC ASPECTS
We turn briefly to some arithmetic aspects of period maps.
6.1. Field of definition of special subvarieties
Once more the geometric case provides us with a motivation and a heuristic. Let f:X rarr Sf: X \rightarrow S be a smooth projective morphism of connected algebraic varieties defined over a number field L subCL \subset \mathbb{C} and let V\mathbb{V} be the natural polarizable ZVHS\mathbb{Z V H S} on S^("an ")S^{\text {an }} with underlying local system R^(∙)f_(**)^("an ")ZR^{\bullet} f_{*}^{\text {an }} \mathbb{Z}. In that case, the Hodge conjecture implies that each special subvariety YY of SS for V\mathbb{V} is defined over bar(Q)\overline{\mathbb{Q}} and that each of the Gal( bar(Q)//L)\operatorname{Gal}(\overline{\mathbb{Q}} / L)-conjugates of YY is again a special subvariety of SS for V\mathbb{V}. More generally, let us say that a polarized ZVHSV=\mathbb{Z} V H S \mathbb{V}=(V_(Z),(V,F^(∙),grad),q)\left(\mathbb{V}_{\mathbb{Z}},\left(\mathcal{V}, F^{\bullet}, \nabla\right), \mathrm{q}\right) on S^(an)S^{\mathrm{an}} is defined over a number field L subCL \subset \mathbb{C} if S,V,F^(∙)S, \mathcal{V}, F^{\bullet} and grad\nabla are defined over LL (with the obvious compatibilities).
Conjecture 6.1. Let V\mathbb{V} be a ZVHS\mathbb{Z} V H S defined over a number field L subCL \subset \mathbb{C}. Then any special subvariety of SS for V\mathbb{V} is defined over bar(Q)\overline{\mathbb{Q}}, and any of its finitely many Gal( bar(Q)//L)\mathrm{Gal}(\overline{\mathbb{Q}} / L)-conjugates is a special subvariety of SS for V\mathbb{V}.
There are only few results in that direction: see [95, THEOREM 0.6] for a proof under a strong geometric assumption; and [81], where it is shown that when SS (not necessarily V\mathbb{V} ) is defined over bar(Q)\overline{\mathbb{Q}}, then a special subvariety of SS for V\mathbb{V} is defined over bar(Q)\overline{\mathbb{Q}} if and only if it contains a bar(Q)\overline{\mathbb{Q}}-point of SS. In [57] Otwinowska, Urbanik, and I provide a simple geometric criterion for a special subvariety of SS for V\mathbb{V} to satisfy Conjecture 6.1. In particular we obtain:
Theorem 6.2 ([57]). Let V\mathbb{V} be a polarized ZVHS\mathbb{Z} V H S on a smooth connected complex quasiprojective variety S. Suppose that the adjoint generic Mumford-Tate group G^("ad ")\mathbf{G}^{\text {ad }} of V\mathbb{V} is simple. If SS is defined over a number field LL, then any maximal (strict) special subvariety Y sub SY \subset S of positive period dimension is defined over bar(Q)\overline{\mathbb{Q}}. If, moreover, V\mathbb{V} is defined over LL then the finitely many Gal( bar(Q)//L)\operatorname{Gal}(\overline{\mathbb{Q}} / L)-translates of YY are special subvarieties of SS for V\mathbb{V}.
As a corollary of Theorems 5.12 and 6.2, one obtains the following, which applies for instance in the situation of Corollary 5.13.
Corollary 6.3. Let V\mathbb{V} be a polarized variation of Z\mathbb{Z}-Hodge structure on a smooth connected quasiprojective variety SS. Suppose that V\mathbb{V} is of level at least 3 , and that it is defined over bar(Q)\overline{\mathbb{Q}}. Then HL(S,V^(ox))_(fpos)\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{fpos}} is an algebraic subvariety of SS, defined over bar(Q)\overline{\mathbb{Q}}.
It is interesting to notice that Conjecture 5.11, which is stronger than Theorem 5.12, predicts the existence of a Hodge generic bar(Q)\overline{\mathbb{Q}}-point in SS for V\mathbb{V} in the situation of Corollary 6.3.
As the criterion given in [57] is purely geometric, it says nothing about fields of definitions of special points. It is, however, strong enough to reduce the first part of Conjecture 6.1 to this particular case:
Theorem 6.4. Special subvarieties for ZVHS\mathbb{Z} V H S defined over bar(Q)\overline{\mathbb{Q}} are defined over bar(Q)\overline{\mathbb{Q}} if and only if it holds true for special points.
6.2. Absolute Hodge locus
Interestingly, Conjecture 6.1 in the geometric case follows from an a priori much weaker conjecture than the Hodge conjecture. Let f:X rarr Sf: X \rightarrow S be a smooth projective morphism of smooth connected complex quasiprojective varieties. For any automorphism sigma in\sigma \inAut(C//Q)\operatorname{Aut}(\mathbb{C} / \mathbb{Q}), we can consider the algebraic family f^(sigma):X^(sigma)rarrS^(sigma)f^{\sigma}: X^{\sigma} \rightarrow S^{\sigma}, where sigma^(-1):S^(sigma)=Sxx_(C,sigma)\sigma^{-1}: S^{\sigma}=S \times_{\mathbb{C}, \sigma}Crarr"∼"S\mathbb{C} \xrightarrow{\sim} S is the natural isomorphism of abstract schemes; and the attached polarizable ZVSH\mathbb{Z V S H}V^(sigma)=(V_(Z)^(sigma),V^(sigma),F^(∙sigma),grad^(sigma))\mathbb{V}^{\sigma}=\left(\mathbb{V}_{\mathbb{Z}}^{\sigma}, \mathcal{V}^{\sigma}, F^{\bullet \sigma}, \nabla^{\sigma}\right) with underlying local system V_(Z)^(sigma)=Rf^(sigma)_(**)^("an ")Z\mathbb{V}_{\mathbb{Z}}^{\sigma}=R f^{\sigma}{ }_{*}^{\text {an }} \mathbb{Z} on (S^(sigma))^("an ")\left(S^{\sigma}\right)^{\text {an }}. The algebraic construction of the algebraic de Rham cohomology provides compatible canonical comparison isomorphisms iota^(sigma):(V^(sigma),F^(∙sigma),grad^(sigma))rarr"∼"sigma^(-1^(**))(V,F^(∙),grad)\iota^{\sigma}:\left(\mathcal{V}^{\sigma}, F^{\bullet \sigma}, \nabla^{\sigma}\right) \xrightarrow{\sim} \sigma^{-1^{*}}\left(\mathcal{V}, F^{\bullet}, \nabla\right) of the associated algebraic filtered vector bundles with connection. More generally, a collection of ZVHS(V^(sigma))_(sigma)\mathbb{Z V H S}\left(\mathbb{V}^{\sigma}\right)_{\sigma} with such compatible comparison isomorphisms is called a (de Rham) motivic variation of Hodge structures on SS, in which case we write V:=V^("Id ")\mathbb{V}:=\mathbb{V}^{\text {Id }}. Following Deligne (see [25] for a nice exposition), an absolute Hodge tensor for such a collection is a Hodge tensor alpha\alpha for V_(s)\mathbb{V}_{s} such that the conjugates sigma^(-1^(**))alpha_(dR)\sigma^{-1^{*}} \alpha_{\mathrm{dR}} of the de Rham component of alpha\alpha defines a Hodge tensor in V_(sigma(s))^(sigma)\mathbb{V}_{\sigma(s)}^{\sigma} for all sigma\sigma. The generic absolute Mumford-Tate group for (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma} is defined in terms of the absolute Hodge tensors as the generic Mumford-Tate group is defined in terms of the Hodge tensors. Thus GsubG^(AH)\mathbf{G} \subset \mathbf{G}^{\mathrm{AH}}. In view of Definition 3.19 the following is natural:
Definition 6.5. Let (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma} be a (de Rham) motivic variation of Hodge structure on a smooth connected complex quasiprojective variety SS. A closed irreducible algebraic subvariety YY of SS is called absolutely special if it is maximal among the closed irreducible algebraic subvarieties ZZ of SS satisfying G_(Z)^(AH)=G_(Y)^(AH)\mathbf{G}_{Z}^{\mathrm{AH}}=\mathbf{G}_{Y}^{\mathrm{AH}}.
In the geometric case, the Hodge conjecture implies, since any automorphism sigma in\sigma \inAut(C//Q)\operatorname{Aut}(\mathbb{C} / \mathbb{Q}) maps algebraic cycles in XX to algebraic cycles on X^(sigma)X^{\sigma}, the following conjecture of Deligne:
Conjecture 6.6 ([33]). Let (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma} be a (de Rham) motivic variation of Hodge structure on S. Then all Hodge tensors are absolute Hodge tensors, i.e., G=G^(AH)\mathbf{G}=\mathbf{G}^{\mathrm{AH}}.
This conjecture immediately implies:
Conjecture 6.7. Let (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma} be a (de Rham) motivic variation of Hodge structure on SS. Then any special subvariety of SS for V\mathbb{V} is absolutely special for (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma}.
Let us say that a (de Rham) motivic variation (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma} is defined over bar(Q)\overline{\mathbb{Q}} if V^(sigma)=\mathbb{V}^{\sigma}=V\mathbb{V} for all sigma in Aut(C// bar(Q))\sigma \in \operatorname{Aut}(\mathbb{C} / \overline{\mathbb{Q}}). In the geometric case, any morphism f:X rarr Sf: X \rightarrow S defined over bar(Q)\overline{\mathbb{Q}} defines such a (de Rham) motivic variation (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma} over bar(Q)\overline{\mathbb{Q}}. Notice that the absolutely
special subvarieties of SS for (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma} are then by their very definition defined over bar(Q)\overline{\mathbb{Q}}, and their Galois conjugates are also special. In particular, Conjecture 6.7 implies Conjecture 6.1 in the geometric case. As proven in [95], Deligne's conjecture is actually equivalent to a much stronger version of Conjecture 6.1, where one replaces the special subvarieties of SS (components of the Hodge locus) with the special subvarieties in the total bundle of V^(ox)\mathcal{V}^{\otimes} (components of the locus of Hodge tensors).
Recently T. Kreutz, using the same geometric argument as in [57], justified Theorem 6.2 by proving:
Theorem 6.8 ([62]). Let (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma} be a (de Rham) motivic variation of Hodge structure on SS. Suppose that the adjoint generic Mumford-Tate group G^("ad ")\mathbf{G}^{\text {ad }} is simple. Then any strict maximal special subvariety Y sub SY \subset S of positive period dimension for V\mathbb{V} is absolutely special.
We refer the reader to [61], as well as [93], for other arithmetic aspects of Hodge loci taking into account not only the de Rham incarnation of absolute Hodge classes but also their â„“\ell-adic components.
ACKNOWLEDGMENTS
I would like to thank Gregorio Baldi, Benjamin Bakker, Yohan Brunebarbe, Jeremy Daniel, Philippe Eyssidieux, Ania Otwinowska, Carlos Simpson, Emmanuel Ullmo, Claire Voisin, and Andrei Yafaev for many interesting discussions on Hodge theory. I also thank Gregorio Baldi, Tobias Kreutz, and Leonardo Lerer for their comments on this text.
FUNDING
This work was partially supported by the ERC Advanced Grant 101020009 "TameHodge."
[3] F. Andreatta, E. Goren, B. Howard, and K. Madapusi-Pera, Faltings heights of abelian varieties with complex multiplication. Ann. of Math. (2) 187 (2018), no. 2, 391-531.
[4] M. Artin, Algebraization of formal moduli. II. Existence of modifications. Ann. of Math. 91 (1970), 88-135.
[5] J. Ax, On Schanuel's conjecture. Ann. of Math. 93 (1971), 1-24.
[6] J. Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups. Amer. J. Math. 94 (1972), 1195-1204.
[7] U. Bader, D. Fisher, N. Miller, and M. Stover, Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds. 2020, arXiv:2006.03008.
[8] W. L. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. 84 (1966), 442-528.
[9] B. Bakker, Y. Brunebarbe, B. Klingler, and J. Tsimerman, Definability of mixed period maps. J. Eur. Math. Soc. (to appear), arXiv:2006.12403.
[10] B. Bakker, Y. Brunebarbe, and J. Tsimerman, o-minimal GAGA and a conjecture of Griffiths. 2019, arXiv:1811.12230.
[11] B. Bakker, B. Klingler, and J. Tsimerman, Tame topology of arithmetic quotients and algebraicity of Hodge loci. J. Amer. Math. Soc. 33 (2020), 917-939.
[12] B. Bakker and J. Tsimerman, The Ax-Schanuel conjecture for variations of Hodge structures. Invent. Math. 217 (2019), no. 1, 77-94.
[13] G. Baldi, B. Klingler, and E. Ullmo, On the distribution of the Hodge locus. 2021, arXiv:2107.08838.
[14] G. Baldi and E. Ullmo, Special subvarieties of non-arithmetic ball quotients and Hodge theory. 2021, arXiv:2005.03524.
[15] E. Bombieri and J. Pila, The number of integral points on arcs and ovals. Duke Math. J. 59 (1989), no. 2, 337-357.
[17] A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. J. Differential Geom. 6 (1972), 543-560.
[18] P. Brosnan and G. Pearlstein, Zero loci of admissible normal functions with torsion singularities. Duke Math. J. 150 (2009), no. 1, 77-100.
[19] P. Brosnan and G. Pearlstein, The zero locus of an admissible normal function. Ann. of Math. (2) 170 (2009), no. 2, 883-897.
[20] P. Brosnan and G. Pearlstein, On the algebraicity of the zero locus of an admissible normal function. Compos. Math. 149 (2013), no. 11, 1913-1962.
[21] P. Brosnan, G. Pearlstein, and C. Schnell, The locus of Hodge classes in an admissible variation of mixed Hodge structure. C. R. Math. Acad. Sci. Paris 348 (2010), no. 11-12, 657-660.
[22] E. Cattani, P. Deligne, and A. Kaplan, On the locus of Hodge classes. J. Amer. Math. Soc. 8 (1995), 483-506.
[23] E. Cattani, A. Kaplan, and W. Schmid, Degeneration of Hodge structures. Ann. of Math. (2) 123 (1986), no. 3, 457-535.
[24] C. L. Chai, Density of members with extra Hodge cycles in a family of Hodge structures. Asian J. Math. 2 (1998), no. 3, 405-418.
[25] F. Charles and C. Schnell, Notes on absolute Hodge classes, Hodge theory. Math. Notes 49 (2014), 469-530.
[26] K. C. T. Chiu, Ax-Schanuel for variations of mixed Hodge structures. 2021, arXiv:2101.10968.
[27] C. Daw and J. Ren, Applications of the hyperbolic Ax-Schanuel conjecture. Compos. Math. 154 (2018), no. 9, 1843-1888.
[39] Z. Gao, Mixed Ax-Schanuel for the universal abelian varieties and some applications. Compos. Math. 156 (2020), no. 11, 2263-2297.
[40] Z. Gao and B. Klingler, The Ax-Schanuel conjecture for variations of mixed Hodge structures. 2021, arXiv:2101.10938.
[41] M. Green, P. Griffiths, and M. Kerr, Mumford-Tate groups and domains. Their geometry and arithmetic. Ann. of Math. Stud. 183, Princeton University Press, 2012.
[42] P. Griffiths, Period of integrals on algebraic manifolds, I. Amer. J. Math. 90 (1968), 568-626.
[43] P. Griffiths, Period of integrals on algebraic manifolds, III. Publ. Math. Inst. Hautes Études Sci. 38 (1970), 125-180.
[44] P. Griffiths, Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems. Bull. Amer. Math. Soc. 76 (1970), 228-296.
[45] P. Griffiths, C. Robles, and D. Toledo, Quotients of non-classical flag domains are not algebraic. Algebr. Geom. 1 (2014), no. 1, 1-13.
[46] P. Griffiths and W. Schmid, Locally homogeneous complex manifolds. Acta Math. 123 (1969), 253-302.
[47] A. Grothendieck, Esquisse d'un programme in Geometric Galois Actions vol. I, London Math. Soc. Lecture Note Ser. 242, Cambridge University Press, 1997.
[48] S. Gukov and C. Vafa, Rational conformal field theories and complex multiplication. Comm. Math. Phys. 246 (2004), no. 1, 181-210.
[52] W. Hodge, Differential forms on a Kähler manifold. Proc. Camb. Philos. Soc. 47 (1951), 504-517.
[53] M. Kashiwara, The asymptotic behavior of a variation of polarized Hodge structure. Publ. Res. Inst. Math. Sci. 21 (1985), no. 4, 853-875.
[54] A. Khovanskii, On a class of systems of transcendental equations. Sov. Math., Dokl. 22 (1980), 762-765.
[55] B. Klingler, Hodge loci and atypical intersections: conjectures. 2017, arXiv:1711.09387, accepted for publication in the book Motives and complex multiplication, Birkhaüser.
[56] B. Klingler and A. Otwinowska, On the closure of the positive dimensional Hodge locus. Invent. Math. 225 (2021), no. 3, 857-883.
[57] B. Klingler, A. Otwinowska, and D. Urbanik, On the fields of definition of Hodge loci. Ann. Éc. Norm. Sup. (to appear), arXiv:2010.03359.
[58] B. Klingler, E. Ullmo, and A. Yafaev, The hyperbolic Ax-Lindemann-Weierstraß conjecture. Publ. Math. Inst. Hautes Études Sci. 123 (2016), no. 1, 333-360.
[60] J. Kóllar and J. Pardon, Algebraic varieties with semialgebraic universal cover. J. Topol. 5 (2012), no. 1, 199-212.
[61] T. Kreutz, â„“\ell-Galois special subvarieties and the Mumford-Tate conjecture. 2021, arXiv:2111.01126.
[62] T. Kreutz, Absolutely special subvarieties and absolute Hodge cycles. 2021, arXiv:2111.00216.
[63] H. B. Mann, On linear relations between roots of unity. Mathematika 12 (1965).
[64] N. Mok, J. Pila, and J. Tsimerman, Ax-Schanuel for Shimura varieties. Ann. of Math. (2) 189 (2019), no. 3, 945-978.
[65] B. Moonen, Linearity properties of Shimura varieties. I. J. Algebraic Geom. 7 (1998), 539-567.
[66] G. Moore, Arithmetic and Attractors. 2007, arXiv:hep-th/9807087.
[67] F. Oort, Canonical liftings and dense sets of CM-points. In Arithmetic Geometry (Cortona, 1994), pp. 228-234, Symp. Math. XXXVII, Cambridge Univ. Press, Cambridge, 1997.
[68] M. Orr, Height bounds and the Siegel property. Algebra Number Theory 12 (2018), no. 2, 455-478.
[69] Y. Peterzil and S. Starchenko, Complex analytic geometry and analytic-geometric categories. J. Reine Angew. Math. 626 (2009), 39-74.
[70] Y. Peterzil and S. Starchenko, Tame complex analysis and o-minimality. In Proceedings of the ICM, Hyderabad, 2010. Available on first author's webpage.
[71] Y. Peterzil and S. Starchenko, Definability of restricted theta functions and families of abelian varieties. Duke Math. J. 162 (2013), 731-765.
[73] J. Pila, O-minimality and diophantine geometry. In Proceedings of the ICM Seoul, 2014.
[74] J. Pila, Functional transcendence via o-minimality. In OO-minimality and diophantine geometry, pp. 66-99, London Math. Soc. Lecture Note Ser. 421, Cambridge Univ. Press, Cambridge, 2015.
[81] M. Saito and C. Schnell, Fields of definition of Hodge loci. In Recent advances in Hodge theory, pp. 275-291, London Math. Soc. Lecture Note Ser. 427, Cambridge Univ. Press, 2016.
[82] W. Schmid, Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22 (1973), 211-319.
[84] H. Shiga and J. Wolfart, Criteria for complex multiplication and transcendence properties of automorphic functions. J. Reine Angew. Math. 463 (1995), 1-25.
[85] S. Tayou, On the equidistribution of some Hodge loci. J. Reine Angew. Math. 762 (2020), 167-194.
[86] S. Tayou and N. Tholozan, Equidistribution of Hodge loci II. 2021, arXiv:2103.15717.
[91] E. Ullmo and A. Yafaev, A characterization of special subvarieties. Mathematika 57 (2011), 263-273.
[92] E. Ullmo and A. Yafaev, Hyperbolic Ax-Lindemann theorem in the cocompact case. Duke Math. J. 163 (2014), 433-463.
[93] D. Urbanik, Absolute Hodge and â„“\ell-adic monodromy. 2020, arXiv:2011.10703.
[94] C. Voisin, Hodge theory and complex algebraic geometry I. Cambridge Stud. Adv. Math. 76, Cambridge University Press, 2002.
[95] C. Voisin, Hodge loci and absolute Hodge classes. Compos. Math. 143 (2007), 945-958945-958.
[96] A. Weil, Abelian varieties and the Hodge ring. In Collected papers III, pp. 421-429, Springer, 1979.
[97] J. Wilkie, Model completeness results for expansions of the ordered field or real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), no. 4, 1051-1094.
[98] X. Yuan and S. Zhang, On the averaged Colmez conjecture. Ann. of Math. (2) 187 (2018), no. 2, 533-638.
[99] U. Zannier, Some problems of unlikely intersections in arithmetic and geometry, with appendices by David Masser. Ann. of Math. Stud. 181, Princeton University Press, 2012.
[100] B. Zilber, Exponential sums equations and the Schanuel conjecture. J. Lond. Math. Soc. (2) 65 (2002), no. 1, 27-44.
CANONICAL KÄHLER METRICS AND STABILITY OF ALGEBRAIC VARIETIES
CHI LI
ABSTRACT
We survey some recent developments in the study of canonical Kähler metrics on algebraic varieties and their relation with stability in algebraic geometry.
The study of canonical Kähler metrics on algebraic varieties is a very active program in complex geometry. It is a common playground of several fields: differential geometry, partial differential equations, pluripotential theory, birational algebraic geometry, and nonArchimedean analysis. We will try to give the reader a tour of this vast program, emphasizing recent developments and highlighting interactions of different concepts and techniques. This article consists of three parts. In the first part, we discuss important classes of canonical Kähler metrics, and explain a well-established variational formalism for studying their existence. In the second part, we discuss algebraic aspects by reviewing recent developments in the study of K-stability with the help of deep tools from algebraic geometry and nonArchimedean analysis. In the third part, we discuss how the previous two parts are connected with each other. In particular, we will discuss the Yau-Tian-Donaldson (YTD) conjecture for canonical Kähler metrics in the first part.
1. CANONICAL KÄHLER METRICS ON ALGEBRAIC VARIETIES
1.1. Constant scalar curvature Kähler metrics
Let XX be an nn-dimensional projective manifold equipped with an ample line bundle LL. By Kodaira's theorem, we have an embedding iota_(m):X rarrP^(N)\iota_{m}: X \rightarrow \mathbb{P}^{N} by using a complete linear system |mL||m L| for m≫1m \gg 1. If we denote by h_(FS)h_{\mathrm{FS}} the standard Fubini-Study metric on the hyperplane bundle over P^(N)\mathbb{P}^{N} with Chern curvature omega_(FS)=-dd^(c)log h_(FS)\omega_{\mathrm{FS}}=-\mathrm{dd}^{\mathrm{c}} \log h_{\mathrm{FS}}, then h_(0)=iota_(m)^(**)h_(FS)^(1//m)h_{0}=\iota_{m}^{*} h_{\mathrm{FS}}^{1 / m} is a smooth Hermitian metric on LL whose Chern curvature omega_(0)=(1)/(m)iota_(m)^(**)omega_(FS)=-dd^(c)log h_(0)\omega_{0}=\frac{1}{m} \iota_{m}^{*} \omega_{\mathrm{FS}}=-\mathrm{dd}^{\mathrm{c}} \log h_{0} is a Kähler form in c_(1)(L)inH^(2)(X,R)c_{1}(L) \in H^{2}(X, \mathbb{R}). In this paper we will use the convention dd^(c)=(sqrt(-1))/(2pi)del bar(del)\mathrm{dd}^{\mathrm{c}}=\frac{\sqrt{-1}}{2 \pi} \partial \bar{\partial}.
We will also use singular Hermitian metrics. An upper-semicontinuous function varphi inL^(1)(omega^(n))\varphi \in L^{1}\left(\omega^{n}\right) is called an omega_(0)\omega_{0}-psh potential if psi+varphi\psi+\varphi is a plurisubharmonic function for any local potential psi\psi of omega_(0)\omega_{0} (i.e., omega_(0)=dd^(c)psi\omega_{0}=\operatorname{dd}^{\mathrm{c}} \psi locally); h_(varphi):=h_(0)e^(-varphi)h_{\varphi}:=h_{0} e^{-\varphi} is then called a psh Hermitian metric on LL. Denote by PSH(omega_(0))\operatorname{PSH}\left(\omega_{0}\right) the space of omega_(0)\omega_{0}-psh functions. By a del bar(del)\partial \bar{\partial}-lemma, any closed positive (1,1)-current in c_(1)(L)c_{1}(L) is of the form omega_(varphi):=omega_(0)+dd^(c)varphi=-dd^(c)log h_(varphi)\omega_{\varphi}:=\omega_{0}+\mathrm{dd}^{\mathrm{c}} \varphi=-\mathrm{dd}^{\mathrm{c}} \log h_{\varphi} with varphi in\varphi \inPSH(omega_(0))\operatorname{PSH}\left(\omega_{0}\right). Moreover, omega_(varphi_(2))=omega_(varphi_(1))\omega_{\varphi_{2}}=\omega_{\varphi_{1}} if and only if varphi_(2)-varphi_(1)\varphi_{2}-\varphi_{1} is a constant. Define the space of smooth strictly omega_(0)\omega_{0}-psh potentials (also called Kähler potentials) by
Fix any varphi inH\varphi \in \mathscr{H}. If omega_(varphi)=sqrt(-1)sum_(i,j)(omega_(varphi))_(ij)dz_(i)^^d bar(z)_(j)\omega_{\varphi}=\sqrt{-1} \sum_{i, j}\left(\omega_{\varphi}\right)_{i j} d z_{i} \wedge d \bar{z}_{j} under a holomorphic coordinate chart, then its Ricci curvature form Ric(omega_(varphi))=(sqrt(-1))/(2pi)sum_(i,j)R_(i bar(j))dz_(i)^^d bar(z)_(j)\operatorname{Ric}\left(\omega_{\varphi}\right)=\frac{\sqrt{-1}}{2 \pi} \sum_{i, j} R_{i \bar{j}} d z_{i} \wedge d \bar{z}_{j} is given by
Then Ric(omega_(varphi))\operatorname{Ric}\left(\omega_{\varphi}\right) is a real closed (1,1)(1,1)-form which represents the cohomology class c_(1)(-K_(X))=c_{1}\left(-K_{X}\right)= : c_(1)(X)c_{1}(X). Here -K_(X)=^^^(n)T^((1,0))X-K_{X}=\wedge^{n} T^{(1,0)} X is the anticanonical line bundle of XX. The scalar curvature of omega_(varphi)\omega_{\varphi} is given by the contraction
Further, omega_(varphi)\omega_{\varphi} is called a constant scalar curvature Kähler (cscK)(\operatorname{cscK}) metric if S(omega_(varphi))S\left(\omega_{\varphi}\right) is the constant S_\underline{S} which is the average scalar curvature and is determined by cohomology classes:
{:(1.2)S_=(n(:c_(1)(X)*c_(1)(L)^(*n-1),[X]:))/(V)quad" with "V=(:c_(1)(L)^(*n),[X]:):}\begin{equation*}
\underline{S}=\frac{n\left\langle c_{1}(X) \cdot c_{1}(L)^{\cdot n-1},[X]\right\rangle}{\mathbf{V}} \quad \text { with } \mathbf{V}=\left\langle c_{1}(L)^{\cdot n},[X]\right\rangle \tag{1.2}
\end{equation*}
The Kähler potential of a cscK metric is a solution to a 4th order nonlinear PDE. In general, there are obstructions to the existence of cscK metrics. For example, the MatsushimaLichnerowicz theorem states that if (X,L)(X, L) admits a cscK metric then the automorphism group Aut(X,L)\operatorname{Aut}(X, L) must be reductive. Our goal is to discuss the Yau-Tian-Donaldson conjecture which would provide a sufficient and necessary algebraic criterion for the existence of cscK\operatorname{cscK} metrics.
1.2. Kähler-Einstein metrics and weighted Kähler-Ricci soliton
Kähler-Einstein metrics form an important class of cscK metrics. A Kähler form omega_(varphi)\omega_{\varphi} is called Kähler-Einstein (KE)(\mathrm{KE}) if Ric(omega_(varphi))=lambdaomega_(varphi)\operatorname{Ric}\left(\omega_{\varphi}\right)=\lambda \omega_{\varphi} for a real constant lambda\lambda. A necessary condition for the existence of KE\mathrm{KE} metrics is that the cohomology class c_(1)(X)inH^(2)(X,R)c_{1}(X) \in H^{2}(X, \mathbb{R}) is either negative, numerically trivial, or positive. The existence for the first two cases was understood in 1970s: there always exists a Kähler-Einstein metric if c_(1)(X)c_{1}(X) is negative (by the work of Aubin and Yau), or if c_(1)(X)c_{1}(X) is numerically trivial (by the work of Yau).
Now we assume that XX is a Fano manifold. In other words, -K_(X)-K_{X} is an ample line bundle, and we set L=-K_(X)L=-K_{X}. Any varphi inH\varphi \in \mathscr{H} corresponds to a volume form
Omega_(varphi):=|s^(**)|_(h_(varphi))^(2)(sqrt(-1))^(n^(2))s^^ bar(s)=Omega_(0)e^(-varphi)quad" with "s=dz_(1)^^cdots^^dz_(n),s^(**)=del_(z_(1))^^cdots^^del_(z_(n))\Omega_{\varphi}:=\left|s^{*}\right|_{h_{\varphi}}^{2}(\sqrt{-1})^{n^{2}} s \wedge \bar{s}=\Omega_{0} e^{-\varphi} \quad \text { with } s=d z_{1} \wedge \cdots \wedge d z_{n}, s^{*}=\partial_{z_{1}} \wedge \cdots \wedge \partial_{z_{n}}
The KE\mathrm{KE} equation in this case is reduced to a complex Monge-Ampère equation for varphi\varphi, namely
We also consider an interesting generalization of Kähler-Einstein metrics on Fano manifolds with torus actions. Assume that T~=(C^(**))^(r)\mathbb{T} \cong\left(\mathbb{C}^{*}\right)^{r} is an algebraic torus and T~=(S^(1))^(r)subTT \cong\left(S^{1}\right)^{r} \subset \mathbb{T} is a compact real subtorus. We will use the following notation:
Assume that T\mathbb{T} acts faithfully on XX. Then there is an induced T\mathbb{T}-action on -K_(X)-K_{X}. Each xi inN_(R)\xi \in N_{\mathbb{R}} corresponds to a holomorphic vector field V_(xi)V_{\xi} on XX. Denote by H^(T)\mathscr{H}^{T} the set of TT-invariant Kähler potentials. For any varphi inH^(T)\varphi \in \mathscr{H}^{T}, the TT-action becomes Hamiltonian with respect to omega_(varphi)\omega_{\varphi}. Denote by m_(varphi):X rarrN_(R)^(**)~=R^(r)\mathbf{m}_{\varphi}: X \rightarrow N_{\mathbb{R}}^{*} \cong \mathbb{R}^{r} the corresponding moment map, and let PP be the image of m_(varphi)\mathbf{m}_{\varphi}. By a theorem of Atiyah-Guillemin-Sternberg, PP is a convex polytope which depends only on the Kähler class c_(1)(L)c_{1}(L). Let g:P rarrRg: P \rightarrow \mathbb{R} be a smooth positive function. The following equation will be called the gg-weighted soliton (or just gg-soliton) equation for varphi inH(-K_(X))^(T)\varphi \in \mathscr{H}\left(-K_{X}\right)^{T}.
An equivalent tensorial equation is given by Ric(omega_(varphi))=omega_(varphi)+dd^(c)log g(m_(varphi))\operatorname{Ric}\left(\omega_{\varphi}\right)=\omega_{\varphi}+\mathrm{dd}^{\mathrm{c}} \log g\left(\mathbf{m}_{\varphi}\right).
Example 1.1. If g(y)=e^(-(:y,xi:))g(y)=e^{-\langle y, \xi\rangle}, then the above equation becomes the standard Kähler-Ricci soliton equation Ric(omega_(varphi))=omega_(varphi)+L_(V_(xi))omega_(varphi)\operatorname{Ric}\left(\omega_{\varphi}\right)=\omega_{\varphi}+\mathscr{L}_{V_{\xi}} \omega_{\varphi} where L\mathscr{L} denotes the Lie derivative.
1.3. Kähler-Einstein metrics on log Fano pairs
Singular algebraic varieties and log pairs are important objects in algebraic geometry, and appear naturally for studying limits of smooth varieties. It is thus natural to study canonical Kähler metric on general log pairs. We recall a definition from birational algebraic geometry. Let XX be a normal projective variety and DD be a Q\mathbb{Q}-Weil divisor. Assume that K_(X)+DK_{X}+D is Q\mathbb{Q}-Cartier. Let mu:Y rarr X\mu: Y \rightarrow X be a resolution of singularities of (X,D)(X, D) with simple normal crossing exceptional divisors sum_(i)E_(i)\sum_{i} E_{i}. We then have an identity
Here A_((X,D))(E_(i)):=a_(i)+1A_{(X, D)}\left(E_{i}\right):=a_{i}+1 is called the log discrepancy of E_(i)E_{i}. The pair (X,D)(X, D) has klt singularities if A_((X,D))(E_(i)) > 0A_{(X, D)}\left(E_{i}\right)>0 for any E_(i)E_{i}. We will always assume that (X,D)(X, D) has klt singularities.
If K_(X)+DK_{X}+D is ample or numerically trivial, Yau and Aubin's existence result had been generalized to the singular and log case in [32], partly based on Kołodziej's pluripotential estimates. There were many related works by Yau, Tian, H. Tsuji, Z. Zhang, and many others.
Now we assume that -(K_(X)+D)-\left(K_{X}+D\right) is ample and call (X,D)(X, D) a log Fano pair. Then one can consider Kähler-Einstein equation or, more generally, gg-soliton equation on (X,D)(X, D). Note that there is a globally defined volume form as in the smooth case: choose a local trivializing section ss of m(K_(X)+D)m\left(K_{X}+D\right) with the dual s^(**)s^{*} and define Omega_(0)=|s^(**)|_(h_(0))^(2//m)(sqrt-1^(mn^(2))s^^:}\Omega_{0}=\left|s^{*}\right|_{h_{0}}^{2 / m}\left(\sqrt{-1}^{m n^{2}} s \wedge\right.bar(s))^(1//m)\bar{s})^{1 / m}. Assume that T\mathbb{T} acts on XX faithfully and preserves the divisor DD. With the notation from before, we say that varphi\varphi is the potential for a gg-weighted soliton on (X,D)(X, D) if varphi\varphi is a bounded omega_(0)\omega_{0}-psh function that satisfies the equation
For any bounded varphi in PSH(omega_(0))\varphi \in \operatorname{PSH}\left(\omega_{0}\right), the gg-weighted Monge-Ampère measure on the left-hand side of (1.5) is well defined by the work of Berman-Witt-Nyström [10] and also by HanLi [38], generalizing the definition of Bedford-Taylor (when g=1g=1 ). It is known that any bounded solution varphi\varphi, if it exists, is orbifold smooth over the orbifold locus of (X,D)(X, D). Moreover, if pp is a regular point of supp(D)\operatorname{supp}(D) such that D=(1-beta){z_(1)=0}D=(1-\beta)\left\{z_{1}=0\right\} locally for a holomorphic function z_(1)z_{1} (with beta in(0,1]\beta \in(0,1] ), then the associated Kähler metric is modeled by C_(beta)xxC^(n-1)\mathbb{C}_{\beta} \times \mathbb{C}^{n-1} where C_(beta)=(C,dr^(2)+beta^(2)r^(2)dtheta^(2))\mathbb{C}_{\beta}=\left(\mathbb{C}, d r^{2}+\beta^{2} r^{2} d \theta^{2}\right) is the 2-dimensional flat cone with cone angle 2pi beta2 \pi \beta.
1.4. Ricci-flat Kähler cone metrics
The class of Ricci-flat Kähler cone metrics is closely related to KE//g\mathrm{KE} / \mathrm{g}-soliton metrics, and is interesting in both complex geometry and mathematical physics (see [57]).
Let Y=Spec(R)Y=\operatorname{Spec}(R) be an (n+1)(n+1)-dimensional normal affine variety with a singularity o in Yo \in Y. Assume that an algebraic torus hat(T)~=(C^(**))^(r+1)\hat{\mathbb{T}} \cong\left(\mathbb{C}^{*}\right)^{r+1} acts faithfully on YY, with oo being the only fixed point. Define hat(N)_(Q), hat(N)_(R)\hat{N}_{\mathbb{Q}}, \hat{N}_{\mathbb{R}} similar to (1.3). The hat(T)\hat{\mathbb{T}}-action corresponds to a weight decomposition of the coordinate ring R=bigoplus_(alpha inZ^(r+1))R_(alpha)R=\bigoplus_{\alpha \in \mathbb{Z}^{r+1}} R_{\alpha}. The Reeb cone can be defined as
hat(N)_(R)^(+)={xi in hat(N)_(R):(:alpha,xi:) > 0" for all "alpha inZ^(r+1)\\{0}" with "R_(alpha)!=0}\hat{N}_{\mathbb{R}}^{+}=\left\{\xi \in \hat{N}_{\mathbb{R}}:\langle\alpha, \xi\rangle>0 \text { for all } \alpha \in \mathbb{Z}^{r+1} \backslash\{0\} \text { with } R_{\alpha} \neq 0\right\}
Any hat(xi)in hat(N)_(R)^(+)\hat{\xi} \in \hat{N}_{\mathbb{R}}^{+}is called a Reeb vector and corresponds to an expanding holomorphic vector field V_( hat(xi))V_{\hat{\xi}}. Assume, furthermore, that YY is Q\mathbb{Q}-Gorenstein and there is a hat(T)\hat{\mathbb{T}}-equivariant nonvanishing section s in|mK_(Y)|s \in\left|m K_{Y}\right|, which induces a hat(T)\hat{\mathbb{T}}-equivariant volume form dV_(Y)=d V_{Y}=(sqrt-1^(m(n+1)^(2))s^^( bar(s)))^(1//m)\left(\sqrt{-1}^{m(n+1)^{2}} s \wedge \bar{s}\right)^{1 / m} on YY. We call the data (Y, hat(xi))(Y, \hat{\xi}) with hat(xi)in hat(N)_(R)^(+)\hat{\xi} \in \hat{N}_{\mathbb{R}}^{+}a polarized Fano cone.
Let hat(T)~=(S^(1))^(r+1)\hat{T} \cong\left(S^{1}\right)^{r+1} be a compact real subtorus of hat(T)\hat{\mathbb{T}}. A hat(T)\hat{T}-invariant function r:Y rarrr: Y \rightarrowR_( >= 0)\mathbb{R}_{\geq 0} is called a radius function for hat(xi)in hat(N)_(R)^(+)\hat{\xi} \in \hat{N}_{\mathbb{R}}^{+}if widehat(omega)=dd^(c)r^(2)\widehat{\omega}=\mathrm{dd}^{\mathrm{c}} r^{2} is a Kähler cone metric on Y^(**)=Y^{*}=Y\\{o}Y \backslash\{o\} and (1)/(2)(rdel_(r)-sqrt(-1)J(rdel_(r)))=V_( hat(xi))\frac{1}{2}\left(r \partial_{r}-\sqrt{-1} J\left(r \partial_{r}\right)\right)=V_{\hat{\xi}}. Here JJ is a complex structure on Y^(**)Y^{*} and widehat(omega)\widehat{\omega} is called a Kähler cone metric if G:=(1)/(2) widehat(omega)(*,J*)G:=\frac{1}{2} \widehat{\omega}(\cdot, J \cdot) on Y^(**)Y^{*} is isometric to dr^(2)+r^(2)G_(S)d r^{2}+r^{2} G_{S} where S={r=1}S=\{r=1\} and G_(S)=G|_(S)G_{S}=\left.G\right|_{S}. In the literature of CR geometry, the induced structure on the link SS by a Kähler cone metric is called a Sasaki structure. Also widehat(omega)=dd^(c)r^(2)\widehat{\omega}=\operatorname{dd}^{\mathrm{c}} r^{2} is called Ricci-flat if Ric( widehat(omega))=0\operatorname{Ric}(\widehat{\omega})=0. In this case, the radius function satisfies the equation (up to rescaling)
If hat(xi)in hat(N)_(Q)\hat{\xi} \in \hat{N}_{\mathbb{Q}}, then widehat(omega)\widehat{\omega} is called quasiregular, and V_( hat(xi))V_{\hat{\xi}} generates a C^(**)\mathbb{C}^{*}-subgroup (: hat(xi):)\langle\hat{\xi}\rangle of hat(T)\hat{\mathbb{T}}. The GIT quotient X=Y////(: hat(xi):)X=Y / /\langle\hat{\xi}\rangle admits an orbifold structure encoded by a log Fano pair (X,D)(X, D). A straightforward calculation shows that a quasiregular (Y, hat(xi))(Y, \hat{\xi}) admits a Ricci-flat Kähler cone metric if and only if (X,D)(X, D) admits a Kähler-Einstein metric.
In general, there are many irregular Ricci-flat Kähler cone metrics, i.e., with hat(xi)in\hat{\xi} \inhat(N)_(R)\\ hat(N)_(Q)\hat{N}_{\mathbb{R}} \backslash \hat{N}_{\mathbb{Q}}. Recent works by Apostolov-Calderbank-Jubert-Lahdili establish an equivalence between Ricci-flat Kähler cone metrics and special gg-soliton metrics. More precisely, fix any hat(chi)in hat(N)_(Q)^(+)\hat{\chi} \in \hat{N}_{\mathbb{Q}}^{+}and consider the quotient (X,D)=Y////(: hat(chi):)(X, D)=Y / /\langle\hat{\chi}\rangle as above. It is shown in [2] (see also [47]) that the Ricci-flat Kähler cone metric on (Y, hat(xi))(Y, \hat{\xi}) is equivalent to the gg-soliton metric on (X,D)(X, D) with g(y)=(n+1+(:y,xi:))^(-n-2)g(y)=(n+1+\langle y, \xi\rangle)^{-n-2} where xi\xi (equivalently, {:V_(xi))\left.V_{\xi}\right) is induced by hat(xi)\hat{\xi} on XX.
1.5. Analytic criteria for the existence
We now review a well-understood criterion for the existence of above canonical Kähler metrics. The general idea is to view corresponding equations as Euler-Lagrange equations of appropriate energy functionals and then use a variational approach to prove that the existence of solutions is equivalent to the coercivity of the energy functionals. First we have the following functionals defined for any varphi inH\varphi \in \mathscr{H} (see (1.1)):
The above H(varphi)\mathbf{H}(\varphi) is usually called the entropy of the measure omega_(varphi)^(n)\omega_{\varphi}^{n}. One can verify that any critical point of M\mathbf{M} is the potential of a cscK metric.
For Kähler-Einstein (KE) metrics on Fano manifolds, we have more functionals:
A critical point of D\mathbf{D} is also a KE\mathrm{KE} potential. These functionals can be generalized to the settings of gg-weighted solitons and Ricci-flat Kähler cone metrics (see [47] for references).
To apply the variational approach, one first needs a "completion" of H\mathscr{H}. Such a completion was defined by Guedj-Zeriahi extending the local study of Cegrell. Following [7], one way to introduce this is to first define the E\mathbf{E} functional for any varphi in PSH(omega_(0))\varphi \in \operatorname{PSH}\left(\omega_{0}\right) by
After the work [6], epsi^(1)\varepsilon^{1} can be endowed with a strong topology which is the coarsest refinement of the weak topology (i.e., the L^(1)L^{1}-topology) that makes E\mathbf{E} continuous. The above energy functionals can be extended to epsi^(1)\varepsilon^{1}, and they satisfy important regularization properties:
Theorem 1.2 (see [6,8][6,8] ). For any varphi inE^(1)\varphi \in \mathcal{E}^{1}, there exists {varphi_(k)}_(k inN)subH\left\{\varphi_{k}\right\}_{k \in \mathbb{N}} \subset \mathscr{H} such that F(varphi_(k))rarrF(varphi)\mathbf{F}\left(\varphi_{k}\right) \rightarrow \mathbf{F}(\varphi) for Fin{E,Lambda,E^(-"Ric "),H}\mathbf{F} \in\left\{\mathbf{E}, \boldsymbol{\Lambda}, \mathbf{E}^{- \text {Ric }}, \mathbf{H}\right\}.
We would like to emphasize the result for F=H\mathbf{F}=\mathbf{H}, which was proved in [8]. The idea of proof there is to first regularize the measure omega_(varphi)^(n)\omega_{\varphi}^{n} with converging entropy and then use Yau's solution to complex Monge-Ampère equations with prescribed volume forms. Later we will encounter the same idea in the non-Archimedean setting.
Another key concept is the geodesic between two finite energy potentials. For varphi_(i)inE^(1),i=0,1\varphi_{i} \in \mathcal{E}^{1}, i=0,1, the geodesic connecting them is the following p_(1)^(**)omega_(0)p_{1}^{*} \omega_{0}-psh function on X xx[0,1]xxS^(1)X \times[0,1] \times S^{1} where p_(1)p_{1} is the projection to the first factor (see [7,26][7,26] ):
{:(1.12)Phi=s u p{Psi:Psi" is "S^(1)"-invariant and "p_(1)^(**)omega_(0)"-psh, "lim_(s rarr i)Psi(*,s) <= varphi(i),i=0,1}:}\begin{equation*}
\Phi=\sup \left\{\Psi: \Psi \text { is } S^{1} \text {-invariant and } p_{1}^{*} \omega_{0} \text {-psh, } \lim _{s \rightarrow i} \Psi(\cdot, s) \leq \varphi(i), i=0,1\right\} \tag{1.12}
\end{equation*}
The concept of geodesics originates from Mabuchi's L^(2)L^{2}-Riemannian metric on H\mathscr{H}. According to the work of Semmes and Donaldson, if varphi_(i)inH,i=0,1\varphi_{i} \in \mathscr{H}, i=0,1, then the geodesic Phi\Phi is a solution to the Dirichlet problem of homogeneous complex Monge-Ampère equation
Since Phi\Phi is S^(1)S^{1}-invariant, we can consider Phi\Phi as a family of omega_(0)\omega_{0}-psh functions {varphi(s)}_(s in[0,1])\{\varphi(s)\}_{s \in[0,1]}.
Theorem 1.3 ( [5,8])[5,8]). Let Phi={varphi(s)}_(s in[0,1])\Phi=\{\varphi(s)\}_{s \in[0,1]} be a geodesic segment in E^(1)\mathcal{E}^{1}. Then (1) s|->s \mapstoE(varphi(s))\mathbf{E}(\varphi(s)) is affine; (2) s|->M(varphi(s))s \mapsto \mathbf{M}(\varphi(s)) is convex.
Results in Theorem 1.3 are important in the variational approach. If a geodesic is smooth, the statements follow from straightforward calculations. However, there are examples (first due to Lempert-Vivas) showing that the solution to (1.13) in general does not have sufficient regularity. So the proofs of the above results are more involved.
In this paper tilde(T)\tilde{\mathbb{T}} will always denote a maximal torus of the linear algebraic group Aut(X,L)\operatorname{Aut}(X, L) and tilde(T)\tilde{T} is a maximal real subtorus of tilde(T)\tilde{\mathbb{T}}. In the following result, we use the translation invariance F(varphi+c)=F(varphi)\mathbf{F}(\varphi+c)=\mathbf{F}(\varphi) for Fin{M,J}\mathbf{F} \in\{\mathbf{M}, \mathbf{J}\} and hence F(omega_(varphi)):=F(varphi)\mathbf{F}\left(\omega_{\varphi}\right):=\mathbf{F}(\varphi) is well defined.
Theorem 1.4 ([9,23,27]). There exists a tilde(T)\tilde{T}-invariant cscK metric in c_(1)(L)c_{1}(L) if and only if M\mathbf{M} is reduced coercive, which means that there exist gamma,C > 0\gamma, C>0 such that for any varphi inH^( tilde(T))\varphi \in \mathscr{H}^{\tilde{T}},
This type of result goes back to Tian's pioneering work in [64] which proves that if XX is a Fano manifold with a discrete automorphism group, then the existence of KählerEinstein metric is equivalent to the properness of the M\mathbf{M}-functional, and is also equivalent to the properness of the D\mathbf{D} functional. Tian's work has since been refined and generalized for other canonical metrics. For the necessity direction (from existence to reduced coercivity), there is now a general principle due to Darvas-Rubinstein ([27]) that can be applied for all previously-mentioned canonical Kähler metrics. The sufficient direction (from reduced coercivity to existence) for Kähler-Einstein metrics is reproved in [6] using pluripotential theory, which works equally well in the setting of log\log Fano pairs. See [10,38][10,38] for the extension to the gg-soliton case. The existence result for smooth cscK metrics is accomplished recently by Chen-Cheng's new estimates [23]. The use of maximal torus appears in [44,45], refining an earlier formulation of Hisamoto [39]. There is also an existence criterion when tilde(T)\tilde{\mathbb{T}} is replaced by any connected reductive subgroup of Aut(X,L)\operatorname{Aut}(X, L) that contains a maximal torus.
2. STABILITY OF ALGEBRAIC VARIETIES AND NON-ARCHIMEDEAN GEOMETRY
2.1. K-stability and non-Archimedean geometry
The concept of K-stability, as first introduced by Tian [64] and Donaldson [30], is motivated by results from geometric analysis. On the other hand, the recent development shows that various tools from algebraic geometry are crucial in unlocking many of its mysteries.
Definition 2.1. A test configuration for a polarized manifold (X,L)(X, L) consists of (X,L)(\mathcal{X}, \mathscr{L}) that satisfies: (i) pi:XrarrC\pi: \mathcal{X} \rightarrow \mathbb{C} is a flat projective morphism from a normal variety X\mathcal{X}, and L\mathscr{L} is a pi\pi semiample Q\mathbb{Q}-line bundle; (ii) There is a C^(**)\mathbb{C}^{*}-action on (X,L)(\mathcal{X}, \mathscr{L}) such that pi\pi is C^(**)\mathbb{C}^{*}-equivariant; (iii) There is a C^(**)\mathbb{C}^{*}-equivariant isomorphism (X,L)xx_(C)C^(**)~=(X xxC^(**),p_(1)^(**)L)(\mathcal{X}, \mathscr{L}) \times_{\mathbb{C}} \mathbb{C}^{*} \cong\left(X \times \mathbb{C}^{*}, p_{1}^{*} L\right).
Configuration (X,L)(\mathcal{X}, \mathscr{L}) is called a product test configuration if there is a C^(**)\mathbb{C}^{*}-equivariant isomorphism (X,L)~=(X xxC,p_(1)^(**)L)(\mathcal{X}, \mathscr{L}) \cong\left(X \times \mathbb{C}, p_{1}^{*} L\right) where the C^(**)\mathbb{C}^{*}-action on the right-hand side is the product action of a C^(**)\mathbb{C}^{*}-action on (X,L)(X, L) with the standard multiplication on C\mathbb{C}.
Two test configurations (X_(i),L_(i)),i=1,2\left(\mathcal{X}_{i}, \mathscr{L}_{i}\right), i=1,2 are called equivalent if there exists a test configuration (X^('),L^('))\left(\mathcal{X}^{\prime}, \mathscr{L}^{\prime}\right) with C^(**)\mathbb{C}^{*}-equivariant birational morphisms rho_(i):X^(')rarrX_(i)\rho_{i}: X^{\prime} \rightarrow X_{i} satisfying rho_(1)^(**)L_(1)=L^(')=rho_(2)^(**)L_(2)\rho_{1}^{*} \mathscr{L}_{1}=\mathscr{L}^{\prime}=\rho_{2}^{*} \mathscr{L}_{2}. For any test configuration (X,L)(\mathcal{X}, \mathscr{L}), by taking fiber product, one can always find an equivalent test configuration (X^('),L^('))\left(\mathcal{X}^{\prime}, \mathscr{L}^{\prime}\right) such that X^(')\mathcal{X}^{\prime} dominates X xxCX \times \mathbb{C}.
Given any test configuration (X,L)(\mathcal{X}, \mathscr{L}), there is a canonical compactification over P^(1)\mathbb{P}^{1} denoted by ( bar(X), bar(L))(\overline{\mathcal{X}}, \overline{\mathscr{L}}) which is obtained by adding a trivial fiber over {oo}=P^(1)\\C\{\infty\}=\mathbb{P}^{1} \backslash \mathbb{C}.
The notion of a test configuration is a way to formulate the degeneration of (X,L)(X, L). In fact, any test configuration is induced by a one-parameter subgroup of PGL(N+1,C)\operatorname{PGL}(N+1, \mathbb{C}) for a Kodaira embedding X rarrP^(N)X \rightarrow \mathbb{P}^{N}.
We will continue our discussion in a framework of non-Archimedean geometry as proposed by Boucksom-Jonsson. Let X^(NA)X^{\mathrm{NA}} denote the Berkovich analytification of XX with respect to the trivial absolute value on C\mathbb{C} (see [18] for references). X^(NA)X^{\mathrm{NA}} is a topological space consisting of real valuations on subvarieties of XX, and contains a dense subset X_(Q)^("div ")X_{\mathbb{Q}}^{\text {div }} consisting of divisorial valuations on XX. Any test configuration (X,L)(\mathcal{X}, \mathscr{L}) defines a function on X^(NA)X^{\mathrm{NA}} in the following way. First, up to equivalence, we can assume that there is a birational morphism rho:XrarrX_(C):=X xxC\rho: \mathcal{X} \rightarrow X_{\mathbb{C}}:=X \times \mathbb{C}. Write L=rho^(**)p_(1)^(**)L+E\mathscr{L}=\rho^{*} p_{1}^{*} L+E where EE is a Q\mathbb{Q}-divisor supported on X_(0)X_{0}. For any v inX^(NA)v \in X^{\mathrm{NA}}, denote by G(v)G(v) the C^(**)\mathbb{C}^{*}-invariant semivaluation on X_(C)X_{\mathbb{C}} that satisfies G(v)|_(C(X))=v\left.G(v)\right|_{\mathbb{C}(X)}=v and G(v)(t)=1G(v)(t)=1 where tt is the coordinate of C\mathbb{C}. One then defines
{:(2.1)phi_((X,L))(v)=G(v)(E)","quad" for any "v inX^(NA):}\begin{equation*}
\phi_{(X, \mathscr{L})}(v)=G(v)(E), \quad \text { for any } v \in X^{\mathrm{NA}} \tag{2.1}
\end{equation*}
The set of such functions on X^(NA)X^{\mathrm{NA}} obtained from test configurations is denoted by H^(NA)\mathscr{H}^{\mathrm{NA}} which is considered as the set of smooth non-Archimedean psh potentials on the analytification of LL. The following functionals, defined on the space of test configurations, correspond to the Archimedean (i.e., complex-analytic) functionals in (1.6)-(1.7):
where we assume that bar(X)\bar{X} dominates X_(P^(1))=X xxP^(1)X_{\mathbb{P}^{1}}=X \times \mathbb{P}^{1} by rho\rho, and L_(P^(1))=p_(1)^(**)L,K_( bar(x)//X_(P1)^(10))^(log)=K bar(x)+L_{\mathbb{P}^{1}}=p_{1}^{*} L, K_{\bar{x} / X_{\mathbb{P} 1}^{10}}^{\log }=K \bar{x}+X_(0)^(red)-(rho^(**)(K_(X xxP^(1))+X xx{0}))X_{0}^{\mathrm{red}}-\left(\rho^{*}\left(K_{X \times \mathbb{P}^{1}}+X \times\{0\}\right)\right). These functionals were defined before the introduction of the non-Archimedean framework. For example, the E^(NA)\mathbf{E}^{\mathrm{NA}} functional appeared in Mumford's study of Chow stability of projective varieties.
Assume that X_(0)=sum_(i)b_(i)F_(i)\mathcal{X}_{0}=\sum_{i} b_{i} F_{i} where F_(i)F_{i} are irreducible components. Set v_(i)=v_{i}=b_(i)^(-1)ord_(F_(i))@p_(1)^(**)inX_(Q)^("div ")b_{i}^{-1} \operatorname{ord}_{F_{i}} \circ p_{1}^{*} \in X_{\mathbb{Q}}^{\text {div }} and let delta_(v_(i))\delta_{v_{i}} be the Dirac measure supported at {v_(i)}\left\{v_{i}\right\}. Chambert-Loir defined the following non-Archimedean Monge-Ampère measure using the intersection theory:
Mixed non-Archimedean Monge-Ampère measures are similarly defined. It then turns out that the functionals from (2.2)-(2.3) can be obtained by using the same formula as in (1.6)-(1.7) but with the ordinary integrals replaced by corresponding non-Archimedean ones, while the H^(NA)\mathbf{H}^{\mathrm{NA}} functional has the following expression after [19]:
Here A_(X)A_{X} is a functional defined on X^(NA)X^{\mathrm{NA}} that generalizes the log discrepancy functional on X_(Q)^("div ")X_{\mathbb{Q}}^{\text {div }} (see [41]). We can now recall the notion of K-stability:
Definition 2.2. A polarized manifold (X,L)(X, L) is K-semistable, K-stable or K-polystable if any nontrivial test configuration (X,L)(\mathcal{X}, \mathscr{L}) for (X,L)(X, L) satisfies M^(NA)(X,L) >= 0,M^(NA)(X,L) > 0\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}) \geq 0, \mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})>0, or M^(NA)(X,L) >= 0\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}) \geq 0 and =0=0 only if (X,L)(\mathcal{X}, \mathscr{L}) is a product test configuration, respectively.
This is like a Hilbert-Mumford's numerical criterion in the Geometric Invariant Theory. ^(1){ }^{1} The recent development of K